What is the formula for the following recursively defined expression

The formula:
1+r+r^2+...+r^n= (r^(n+1)-1) / (r-1)
is true for all real numbers r except r=1 and for all integers n >=0. Use this fact to solve this problem:


If n is an integer and n >= 1, find a formula for the following expression:
2^(n)-2^(n-1)+2^(n-2)-2^(n-3)+...+ -1^(n-1)*2 +(-1)^n

I am completely lost on this problem and need a detailed solution ASAP. Thanks!

Factor out 2ⁿ.

2ⁿ - 2^(n-1) + 2^(n-2) - .... + (-1)^(n-1)2 + (-1)ⁿ =

2ⁿ[1 - 2^(-1) + 2^(-2) - ... + (-1)^(n-1) 2^(1-n) + (-1)ⁿ 2^(-n)]=

2ⁿ[1 + (-1)/2 + (-1)²/2² + (-1)^3/2^3 + .... + (-1)^(n-1)/2^(n-1) + (-1)ⁿ/2ⁿ]

So r = -1/2.

Your sum is

2ⁿ((-1/2)^(n+1) - 1)/(-1/2 - 1) = (2ⁿ - 1)/3

Your series is a GP with common ratio - 1/2, so the sum is

(( - 1/2 )^(n + 1 ) - 1) /(-(1/2) - 1) , which you can easily simplify.