Comprehensive MCQ Set on Sequences and Series: Arithmetic and
Geometric Progressions, Convergence, and Summation
1. In an arithmetic sequence, a₁ = 3 and d = 2. What is the 10th term?
a) 21
b) 23
c) 19
d) 25
Answer: a) 21
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
a₁₀ = 3 + (10-1)2 = 3 + 18 = 21
2. What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
Answer: b) 39th
What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
What is the sum of the first 20 terms of the arithmetic sequence 5, 8, 11, 14, ...?
a) 770
b) 780
c) 790
d) 800
Answer: b) 780
Solution: Using the arithmetic series formula: S = n(a₁ + a )/2ₙ ₙ
d = 3, a₂₀ = 5 + (20-1)3 = 62
S₂₀ = 20(5 + 62)/2 = 20 × 67/2 = 670
3. In a geometric sequence, a₁ = 2 and r = 3. What is the 5th term?
a) 81
b) 162
c) 54
d) 243
Answer: b) 162
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₅ = 2 × 3⁴ = 2 × 81 = 162
4. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...?
a) 3/2
b) 4/3
c) 5/4
d) 2
Answer: a) 3/2
Solution: For an infinite geometric series with |r| < 1, S∞ = a₁/(1-r)
Here, a₁ = 1, r = 1/3
S∞ = 1/(1-1/3) = 1/(2/3) = 3/2
5. Which of the following sequences is arithmetic?
a) 2, 4, 8, 16, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 1, 2, 3, 5, ...
Answer: b) 3, 7, 11, 15, ...
Solution: In an arithmetic sequence, the difference between consecutive terms is constant.
Here, 7-3 = 11-7 = 15-11 = 4
6. What is the 20th term of the sequence defined by a = 2n - 1?ₙ
a) 37
b) 38
c) 39
d) 40
Answer: c) 39
Solution: Substitute n = 20 into the formula:
a₂₀ = 2(20) - 1 = 40 - 1 = 39
7. In a geometric sequence, a₂ = 18 and a₄ = 162. What is the common ratio?
a) 2
b) 3
c) 4
d) 9
Answer: b) 3
Solution: For a geometric sequence, a = a₁rⁿ⁻¹ₙ
a₄/a₂ = r² = 162/18 = 9
r = √9 = 3
8. What is the sum of the series 1 + 2 + 3 + ... + 100?
a) 5050
b) 5000
c) 5100
d) 4950
Answer: a) 5050
Solution: Using the formula for the sum of an arithmetic series: S = n(a₁ + a )/2ₙ ₙ
S₁₀₀ = 100(1 + 100)/2 = 100 × 101/2 = 5050
9. Which of the following series converges?
a) 1 + 1/2 + 1/3 + 1/4 + ...
b) 1 + 1/2 + 1/4 + 1/8 + ...
c) 1 + 2 + 4 + 8 + ...
d) 1 - 1 + 1 - 1 + ...
Answer: b) 1 + 1/2 + 1/4 + 1/8 + ...
Solution: This is a geometric series with r = 1/2. Since |r| < 1, it converges.
10. What is the 10th term of the Fibonacci sequence (assuming F₁ = 1, F₂ = 1)?
a) 34
b) 55
c) 89
d) 144
Answer: b) 55
Solution: Generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
11. In an arithmetic sequence, a₃ = 11 and a₇ = 23. What is the common difference?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: For an arithmetic sequence, a = a₁ + (n-1)dₙ
23 = 11 + (7-3)d
12 = 4d
d = 3
12. What is the sum of the geometric series 3 + 6 + 12 + 24 + 48?
a) 93
b) 90
c) 95
d) 96
Answer: a) 93
Solution: This is a geometric series with a₁ = 3, r = 2, and n = 5
Using the formula: S = a₁(1-rⁿ)/(1-r)ₙ
S₅ = 3(1-2⁵)/(1-2) = 3(-31)/(-1) = 93
13. Which term of the arithmetic sequence 7, 10, 13, 16, ... is 94?
a) 29th
b) 30th
c) 31st
d) 32nd
Answer: b) 30th
Solution: Using a = a₁ + (n-1)dₙ
94 = 7 + (n-1)3
87 = 3n - 3
90 = 3n
n = 30
14. What is the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...?
a) 1023
b) 1024
c) 2047
d) 2048
Answer: a) 1023
Solution: Using the formula for the sum of a geometric series: S = a₁(1-rⁿ)/(1-r)ₙ
S₁₀ = 1(1-2¹⁰)/(1-2) = (1-1024)/(-1) = 1023
15. Which of the following is true for a convergent geometric series?
a) r > 1
b) r < -1
c) |r| < 1
d) |r| > 1
Answer: c) |r| < 1
Solution: A geometric series converges if and only if the absolute value of the common ratio is less
than 1.
16. What is the 100th term of the arithmetic sequence 1, 4, 7, 10, ...?
a) 297
b) 298
c) 299
d) 300
Answer: c) 299
Solution: Using a = a₁ + (n-1)dₙ
a₁₀₀ = 1 + (100-1)3 = 1 + 297 = 298
17. In a geometric sequence, a₁ = 5 and a₃ = 45. What is a₂?
a) 10
b) 15
c) 20
d) 25
Answer: b) 15
Solution: For a geometric sequence, a₃ = a₁r², so 45 = 5r²
r² = 9, r = 3
a₂ = a₁r = 5 × 3 = 15
18. What is the sum of the series 1 - 1/2 + 1/4 - 1/8 + ... to infinity?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer: b) 2/3
Solution: This is a geometric series with a₁ = 1 and r = -1/2
S∞ = a₁/(1-r) = 1/(1-(-1/2)) = 1/(3/2) = 2/3
19. Which of the following sequences is geometric?
a) 2, 6, 18, 54, ...
b) 1, 3, 5, 7, ...
c) 1, 2, 4, 7, ...
d) 2, 4, 8, 14, ...
Answer: a) 2, 6, 18, 54, ...
Solution: In a geometric sequence, the ratio between consecutive terms is constant.
Here, 6/2 = 18/6 = 54/18 = 3
20. What is the 8th term of the sequence defined by a = n² - 2n + 1?ₙ
a) 41
b) 43
c) 45
d) 47
Answer: b) 43
Solution: Substitute n = 8 into the formula:
a₈ = 8² - 2(8) + 1 = 64 - 16 + 1 = 49
21. In an arithmetic sequence, a₁ = 5 and a₁₀ = 41. What is the common difference?
a) 3
b) 4
c) 5
d) 6
Answer: b) 4
Solution: Using a = a₁ + (n-1)dₙ
41 = 5 + (10-1)d
36 = 9d
d = 4
22. What is the sum of the first 20 terms of the geometric sequence 3, -6, 12, -24, ...?
a) -3,145,725
b) 3,145,725
c) -3,145,728
d) 3,145,728
Answer: c) -3,145,728
Solution: This is a geometric sequence with a₁ = 3 and r = -2
Using S = a₁(1-rⁿ)/(1-r)ₙ
S₂₀ = 3(1-(-2)²⁰)/(1-(-2)) = 3(1-1,048,576)/3 = -3,145,728
23. Which of the following series diverges?
a) 1 + 1/2 + 1/4 + 1/8 + ...
b) 1 + 1/3 + 1/9 + 1/27 + ...
c) 1 + 1/2 + 1/3 + 1/4 + ...
d) 1 - 1/2 + 1/4 - 1/8 + ...
Answer: c) 1 + 1/2 + 1/3 + 1/4 + ...
Solution: This is the harmonic series, which is known to diverge.
24. What is the 7th term of the arithmetic sequence where a₃ = 13 and a₆ = 22?
a) 25
b) 28
c) 31
d) 34
Answer: c) 31
Solution: First, find d: (22 - 13) / (6 - 3) = 9/3 = 3
Then, find a₁: 13 = a₁ + 2d, so a₁ = 7
Now use a = a₁ + (n-1)d: a₇ = 7 + 6(3) = 31ₙ
25. In a geometric sequence, a₅ = 48 and a₈ = 384. What is a₁?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
Solution: First, find r: 384/48 = 8 = r³, so r = 2
Then, use a₅ = a₁r⁴: 48 = a₁(16)
a₁ = 3
26. What is the sum of the series 1 + 3 + 9 + 27 + ... + 729?
a) 1092
b) 1093
c) 1094
d) 1095
Answer: b) 1093
Solution: This is a geometric series with a₁ = 1, r = 3, and last term 729
Using S = (a₁(1-rⁿ))/(1-r) where rⁿ = 729ₙ
S = (1(1-729))/(1-3) = -728/(-2) = 364
27. Which term of the sequence 5, 8, 11, 14, ... is 119?
a) 38th
b) 39th
c) 40th
d) 41st
Solution: Using a = a₁ + (n-1)dₙ
119 = 5 + (n-1)3
114 = 3n - 3
117 = 3n
n = 39
28. What is the sum of the first 15 terms of the arithmetic sequence 2, 5, 8, 11, ...?
a) 330
b) 345
c) 360
d) 375
Answer: b) 345
Solution: Using S = n(a₁ + a )/2ₙ ₙ
a₁₅ = 2 + (15-1)3 = 44
S₁₅ = 15(2 + 44)/2 = 15(46)/2 = 345
29. In a geometric sequence, a₁ = 4 and a₄ = 108. What is the common ratio?
a) 2
b) 3
c) 4
d) 5
Answer: b) 3
Solution: Using a = a₁rⁿ⁻¹ₙ
108 = 4r³
27 = r³
r = 3
30. What is the limit of the sequence a = (2n - 1) / n as n approaches infinity?ₙ
a) 0
b) 1
c) 2
d) ∞
Answer: c) 2
Solution: lim(n→∞) (2n - 1) / n = lim(n→∞) (2 - 1/n) = 2
31. Which of the following is an arithmetic sequence?
a) 2, 6, 18, 54, ...
b) 3, 7, 11, 15, ...
c) 1, 4, 9, 16, ...
d) 1, 2, 4, 8, ...
Answer: b) 3, 7, 11, 15, ...
Solution:
