Easy proof by induction (1 Viewer)

[Dota55]

hi yeah um...im kinda stuck on the third step of this question.

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

Thanks in advance!

 

[Trebla]

Well first of all it should be:
1 + 2 + 4 + ..... + 2^{n - 1} = 2ⁿ - 1

Step 1 - Obvious...
LHS = 1
RHS = 2 - 1
= 1
LHS = RHS, hence true for n = 1

Step 2 - Assume the statement is true for n = k
1 + 2 + 4 + ..... + 2^{k - 1} = 2^k - 1

Step 3 - Prove it is true for n = k + 1
1 + 2 + 4 + ..... + 2^k = 2^{k + 1} - 1

LHS = 1 + 2 + 4 + ..... + 2^k
= [1 + 2 + 4 + ..... + 2^{k - 1}] + 2^k
= 2^k - 1 + 2^k by assumption
= 2.2^k - 1
= 2^{k + 1} - 1
= RHS

Then insert conclusion...

 

[powerdrive]

thought u had to show that it is true for n=1 sumwhere in there.
learnt this at the beginning of last year and cant remember much.

yeh, it's step 1, he said it was obvious.