Suppose an/n+1 +...+a0/1=0.
Prove f(x) =anxn +...+a0 has a root between zero and one.
I'm pretty sure this is induction, but I'm not completely sure.
Mean Value Theorem probably
The Attempt at a Solution
Well f(0)=a0 and f(1)=an + ... + a0.
If it is induction, it is easy to show that when n=0, f(x)=0 for all x and thus it is obviously constant and has infinitely many roots between 0 and 1.
Assuming a root between 0 and 1 for n=k, the case for n=k+1 just doesn't seem very manipulable to me to create any kind of cancellation or truth arising from n=k. I'm pretty sure I have to manipulate the an/n+1 +...+a0=0 deal, but how to do that I have no clue.
Also, should I aim to prove that f(0) and f(1) have opposite signs? This gives me the end result immediately from the Mean Value Theorem. How would I accomplish that? What if they are both zero?
Is induction the right method here, or am I barking up the wrong tree?
Any help would be much appreciated.