Math Homework

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Section 1-2

An Application of Inductive Reasoning:

Number Patterns

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Number Sequences

Number Sequence A list of numbers having a first number, a second number, and so on, called the terms of the sequence. Arithmetic Sequence A sequence that has a common difference between successive terms. (add/subtract same number)

16 20 24 28 …

+4 +4 +4 Geometric Sequence A sequence that has a common ratio between successive terms. (multiply (*)/divide by same number) 3 12 48 192 …

*4 *4 *4

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14, 22, 32, 44,...

14 22 32 44

8 10 12 Find differences

2 2 Find differences

Example: Successive Differences

Process to determine the next term of a sequence

using subtraction to find a common difference.

Use the method of successive differences to find the

next number in the sequence.

Build up to next term: 58

2

14

58

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Number Patterns and Sum Formulas

Sum of the First n Odd Counting Numbers

If n is any counting number, then

Ex: 1 + 2 + 5+…+ 51 = (Hint: to find n, add 1 to last term and divide

by 2: n = (51+1)/2 = 26) S = n2 = 262 = 676

Sum of the First n Counting Numbers For any counting number n,

( 1) and 1 2 3 .

2

n n n

     

2 1 3 5 (2 1) .     n n

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Example: Sum Formula

Use a sum formula to find the sum

1 2 3 48.   

Solution ( 1)

1 2 3 2

n n n

     

with n = 48:

48(48 1) 1176.

2

 

Use the formula

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Figurate Numbers

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Formulas for Triangular, Square, and

Pentagonal Numbers

For any natural number n, ( 1)

the th triangular number is given by , 2

n

n n n T

 

2 the th square number is given by , andnn S n

(3 1) the th pentagonal number is given by .

2

 n

n n n P

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Example: Figurate Numbers

Use a formula to find the sixth pentagonal number

Solution (3 1)

2 n

n n P

 

with n = 6:

6

6[6(3) 1] 51.

2 P

  

Use the formula