2. a) Show by mathematical induction that
1/2 + 2/2^2 + 3/2^3 + ... + n/2^n = 2 - n+2/2^n
My work:
1. Show p(1) true. I did this part.
2. Assume p(k) true => assume p(k+1) true
I'm a bit stuck here. Here's how I set it up
(2 - k+2 / 2^k) + (k+1 / 2^k+1) = (2 - k+2 / 2^k) + (k+1 / 2^k+1)
I think all I have to do is simplify the right side so that it equals 2 - k+3 / 2^k+1. Please help.
To do an induction proof, you assume the n = k step, and then try to show the n = k + 1 step. You don't assume the conclusion; you have to prove it.
I'm not sure what you're doing with the n = k + 1 step...?
Since the previous (assumption) step was for n = k, you have assumed the following:
You are now needing to work with:
...and you hope to manipulate the above to be in the form:
The first proof step is (usually) to substitute from the assumption step. In this case, we get:
A good first step might be to convert the second term to a common denominator with the third term, noting that .