Here is the question:
a_0 + a_1/2 + ... + a_n/(n+1) = 0, all a's are real.
Show that at least one real root betw. 0 and 1 exists for a_0 + a_1*x + ... + a_n*x^n = 0.
I decided to go at this with induction. The n=1 step is easy; just solve a 2X2 system of equations. But I am stuck on the inductive step. I asked a classmate, and he suggested integrating the polynomial and then using the intermediate value theorem. But integrating the polynomial with respect to x doesn't produce anything that seems relevant or usable. What should I do?