Question

I got this problem from an old algebra book and I seem to have got a mental block with Part 2:
Part 1. Find the nth term for the sequence 1, 3/2, 5/4, 7/8.......?
Part 2. If the sum of n terms is denoted by s, write down the series obtained by subtracting 1/2s from s, and find its value in terms of n. Hence find s.
I obtained nth term = (2n - 1)/2^(n-1) for Part 1.

Answer 1

Step-by-step explanation:

S = 1 + 3/2 + 5/4 + 7/8 + ...

½S = 1/2 + 3/4 + 5/8 + 7/16 + ...

Subtracting:

S − ½S = 1 + 3/2 − 1/2 + 5/4 − 3/4 + 7/8 − 5/8 + ...

Simplifying:

S − ½S = 1 + 1 + 1/2 + 1/4 + ...

Writing in summation form:

S − ½S = 1 + ∑ ½ⁿ⁻¹

½S = 1 + ∑ ½ⁿ⁻¹

S = 2 + 2 ∑ ½ⁿ⁻¹

Using formula for sum of a geometric series:

S = 2 + 2 (1 / (1 − ½))

S = 2 + 2 (2)

S = 6