Question

For all integers n ≥ 1, prove the following statement using mathematical induction. 1+2^1 +2^2 +...+2^n = 2^(n+1) −1

Here's what I have so far
1. Prove the base step

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

Someone else suggested that the base step is :
1+2^1=2^(1+1)-1
This returns a true value of 3=3 but doesn't work for any other n... Can some one tell me what the base step is? and why?

2. State the inductive hypothesis.

Assume the statement is also true for k

3. State what you have to show.

4. Proof proper:

Answer 1

sure it works.
1 = 2^1-1
1+2 = 2^2-1
1+2+4 = 2^3-1
...
So, the base step works fine.

Assume it's true for n=k. Now, if n=k+1 we have

1+2+4+...+2^k+2^(k+1) = 2^(k+1)-1 + 2^(k+1)
= 2*2^(k+1)-1
= 2^(k+2)-1
= 2^((k+1)+1)-1

QED

Answer 2

2+4+6+....+2n=n(n+1)