I'm trying to prove the statement $$ \sum_{\ell\text{ odd}}^{n} \binom{n}{\ell} = 2^{n-1} $$ by induction. The base case is $$\sum_{\ell\text{ odd}}^{1} \binom{n}{\ell} = \binom{1}{1} = 2^{1-1} = 1.$$
For the induction step, I'm assuming that $$\sum_{\ell\text{ odd}}^{n} \binom{n}{\ell} = 2^{n-1}$$ and trying to prove $$\sum_{\ell\text{ odd}}^{n+1} \binom{n+1}{\ell} = 2^{n+1-1} = 2^{n}$$
Because $$ \binom{n+1}{k} = \binom{n}{k} + \binom{n}{k-1}$$
$$\sum_{\ell\text{ odd}}^{n+1} \binom{n+1}{\ell} = \sum_{\ell\text{ odd}}^{n+1} \binom{n}{\ell} + \sum_{\ell\text{ odd}}^{n+1} \binom{n}{\ell-1}.$$
The first part of the expression is equal to $$2^{n-1}$$ so I'm just trying to prove that $$\sum_{\ell\text{ odd}}^{n+1} \binom{n}{\ell-1} = \binom{n}{0} + \binom{n}{2} + \dotsb = 2^{n-1}$$ but I'm stuck at this step.