Binomial Theorem Question about a Proof

Question

From the textbook:

Suppose $(1+x+x^2+...+x^k)^n = a_0 + a_1x+a_2x^2 + ... + a_{kn}x^{kn}$.

Here is the question I'm working on:

Show that $a_0 + a_1 + a_2 + ... + a_{kn} = (k+1)^n$.

I know I need to use the Binomial Theorem in some way because I can expand $(k+1)^n$ this as

$(k+1)^n$ = ${n}\choose{0}$ $k^n$ + ${n}\choose{1}$ $k^{n-1}$ + ...

How can I use this expansion, if done correctly, to get started on the proof?

Answer 1

You do not need the binomial theorem. Just put in $x=1$. Note that you get $$a_0 + a_1 + a_2 + ... + a_{kn} = (\underbrace{1+1+\dots+1}_{k+1 \textrm{ times}})^n=(k+1)^n$$