Question

2 + 2² + 2³ + .... + 2^n = 2 ( 2^n - 1 )​

Answer 1

Step-by-step explanation:

n2<2n

P(n) is the statement n2<2n

Claim: For all n>k, where k is any integer, P(n)

(since k is any integer, I assume I have to prove this for positive and negative integers)

So let base case be P(1), and I have to prove P(n) for n≥1 and n<1

P(1) is 12<21 which is clearly true.

Induction hypothesis: k2<2k

Inductive step (k+1)2<2(k+1)

(k+1)2=k2+2k+1

k2+2k+1<2k+2k+1 by inductive hypothesis

Not sure how to proceed. Is my previous intuition that I have to prove this for n≥1 and n<1 correct?

Answer 2

Answer:

That's true

Step-by-step explanation:

In rhs we take 2 as common then it will be

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

2(2n-1)=2(2n-1)