Let c(n) be the constant term in the expansion of
(x + 9)n.
prove by induction that
c(n) = 9n
for all n is in n.
(induction on n.) the constant term of
(x + 9)1
is

= 9
.

suppose as inductive hypothesis that the constant term of
(x + 9)k − 1
is

for some
k > 1.

then
(x + 9)k = (x + 9)k − 1 ·

,
so its constant term is

· 9 =
,
as required.

Presumably you meant to write



For , we have




Suppose the claim holds for , i.e. that



Then for , we have



Every term in the expansion of the first term will have degree at least 1 ( at the most and at the least), so we can safely ignore these terms.

This leaves us with



We already know the constant term of the expansion here is . Multiplying by 9, we then are left with , proving the claim.