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FALL TERM, PRECALCULUS, HOMEWORK #10

1. Recall, a sequence is a function f : N → S where S is an arbitrary set. Let S = E be the set of even numbers. Prove or give a counterexample: There exists an explicit bijection N

ϕ →E.

2. Let the n-th term of a recursive sequence be given by an = 2an−1 + 3an−2. Suppose that a4 = −1 and a5 = 1. Find a7.

3. Let the n-th term of a sequence have the formula an = (−1)n 2n

. Find the n-th partial sum, Sn, for n = 1, 2, 3. Is this sequence an example of an alternating sequence?

4. Evaluate the sum: 5∑

n=1

1

n −

1

n + 1

5. An infinite series is given by 1 − 1 3

+ 1 9 − 1

27 + · · ·− . Write this series in Σ notation.

6. Prove the following properties of sums:

n∑ k=1

(ak − bk) = n∑

k=1

ak − n∑

k=1

bk.

7. The 12th term of an arithmetic sequence is 32, and the fifth term is 18. Find the 20th term.

8. The first term of a geometric sequence is 8, and the second term is 4. Find the fifth term.

9. Find the sum of the infinite series: ∞∑ n=1

( 3

4

)n

10. Express the repeating decimal as a fraction: 0.253