Question

Need help with a mathematical induction problem.
1+2+2^2+2^3...+2^(n-1) =2^(n) -1.

I keep getting down to 4^(k-1) but It doesn't look like the answer. I plugged in (k+1) into 2^(n) -1 to know what I'm supposed to end up with to prove this formula works for the k+1 term.

Answer 1

Does it start?

1 + 2 + 4 = 7
2^3 - 1 = 8 - 1 = 7

Does one imply the next?

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

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

And we appear to be done. Why do you have a four (4) in that base?

Answer 2

I don't understand how you simplified from the last step. 2^(n) -1 + 2^n = 2(2^n) - 1 = 2^(n+1) - 1
I added the two 2^(k) and then put -1 to get 4^(K) -1

Answer 3

?? \(2^{n} + 2^{n} = 2^{n}(1+1) = 2\cdot 2^{n} = 2^{n+1}\)

\(House + House = 2*House \ne Duplex\)

Why would you add or multiply the bases?