Let k be a natural number, and r a real number with |r|<1. Prove (by induction on k) that for any polynomial P of degree k, there is a polynomial Q of degree k with Q(n+1)r^(n+1) - Q(n)r^n = P(n)r^n
Hint: consider differences of successive terms for (n^k)(r^n) and use the inductive hypothesis
The Attempt at a Solution
Base case: degree 0
a_0 r^n = b_0 r^(n+1) - b_0 r^n
Is it ok to just rearrange this, and find b_0 in terms of a_n, r^n etc..
a_0/(r-1) = b_0
So for some polynomial of degree 0 I can always find another using the fact that b_0 = a_0/(r-1). However, when I try to do this with higher degrees, I get the coefficients in terms of n which means that the polynomial depends on what n is which means I'm not proving I can find one that satisfies for all n. Any ideas?