How can I compute this sum of binomial

Question

Is there any way to compute the following sum: $\displaystyle{ \sum_{\ell = {n + 1 \over{\vphantom{\LARGE A}2}}}^{n}{n \choose \ell}5^{n - \ell}}$ where $n$ is odd. Thank you.

Answer 1

Hint : try to see the connection with newton's binomium theorem.

How do we rewrite $5^{n+1}$ with the binomial coëfficiënts ?

Use the symmetry of the binomial coefficients.

Answer 2

Hints : What can you say about the connection $\displaystyle{ {n \choose \ell}}$ with $\displaystyle{ {\ell-n \choose \ell}}$ ?

Look at $a_0 + a_1 x + a_2 x^2+...a_n x^n$

What happens if you multiply that with $(x^n+1)$ ?