Question

Let C(n) be the constant term in the expansion of

Answer 1

Presumably you meant to write

$C(n)=9^n$

For $n=1$, we have

$(x+9)^1=x+9$
$C(1)=9^1=9$

Suppose the claim holds for $n=k$, i.e. that

$C(k)=9^k$

Then for $n=k+1$, we have

$(x+9)^{k+1}=(x+9)^k(x+9)=x(x+9)^k+9(x+9)^k$

Every term in the expansion of the first term will have degree at least 1 ($x^{k+1}$ at the most and $9^kx$ at the least), so we can safely ignore these terms.

This leaves us with

$9(x+9)^k$

We already know the constant term of the expansion here is $C(k)=9^k$. Multiplying by 9, we then are left with $C(k+1)=9^{k+1}$, proving the claim.