Presumably you meant to write
![C(n)=9^n](/tpl/images/0388/4283/e5030.png)
For
![n=1](/tpl/images/0388/4283/b4047.png)
, we have
![(x+9)^1=x+9](/tpl/images/0388/4283/65d0a.png)
![C(1)=9^1=9](/tpl/images/0388/4283/cbe0a.png)
Suppose the claim holds for
![n=k](/tpl/images/0388/4283/7386c.png)
, i.e. that
![C(k)=9^k](/tpl/images/0388/4283/0270b.png)
Then for
![n=k+1](/tpl/images/0388/4283/fc83f.png)
, we have
![(x+9)^{k+1}=(x+9)^k(x+9)=x(x+9)^k+9(x+9)^k](/tpl/images/0388/4283/24356.png)
Every term in the expansion of the first term will have degree at least 1 (
![x^{k+1}](/tpl/images/0388/4283/d0fdf.png)
at the most and
![9^kx](/tpl/images/0388/4283/951ba.png)
at the least), so we can safely ignore these terms.
This leaves us with
![9(x+9)^k](/tpl/images/0388/4283/c0be1.png)
We already know the constant term of the expansion here is
![C(k)=9^k](/tpl/images/0388/4283/0270b.png)
. Multiplying by 9, we then are left with
![C(k+1)=9^{k+1}](/tpl/images/0388/4283/85f2c.png)
, proving the claim.