You need to show:
So the LHS is your original sum and the next cubic term, the RHS is
the same form as your expression, but for k+1
I want to prove by induction for all positive integers n but I'm having a bit of trouble after a while:
for n = 1: = so the statement's true for n = 1.
assume for n = k: =
and for n = k = 1:
I'm not sure how to make progress from here. Is there anyone who can help out? Thanks
Let us assume that, forTherefore:
Then:So when is also true.
Since you have already shown that this is true for , this is all you need to complete the proof.