Question

Help with proving by induction?

Answer 1

\[2(1+3+3^{2}+3^{3}+...+3^{k-1}=3^{k}-1\]

Answer 2

what i have so far is \[2(1+3+3^{2}+3^{3}+...+3^{k-1}+3^{k})=3^{k}-1+2(3^{k})\] but i'm not sure how to finish it.

Answer 3

\(2(1+3+3^{2}+3^{3}+...+3^{k-1}) = 3^{k}-1\)
Prove the base case when k = 1. When k = 1,
LHS = \(2(1) = 2\)
RHS = \(3^1-1 = 3-1 = 2\)
So it is true for k = 1.

Assume the identity is true for k.
Then for (k+1) terms:
\(2(1+3+3^{2}+3^{3}+...+3^{k-1} + 3^k) = 3^k - 1 + 2*3^k = 3^k(2+1)-1 = \\
3^k*3 - 1 = 3^{k+1} - 1\).

So if it is true for k, it is true for k+1.

Answer 4

Thank you so much!

Answer 5

You are welcome.