Question 24
Let $P_{k}(x) = 1 + x + x^{2} + \cdots + x^{k-1}$. Show that
$$
\sum_{k=1}^{n} \binom{n}{k} P_{k}(x) = 2^{n-1} P_{n}\left(\frac{1+x}{2}\right)
$$
for every real number $x$ and every positive integer $n$.
$$
\sum_{k=1}^{n} \binom{n}{k} P_{k}(x) = 2^{n-1} P_{n}\left(\frac{1+x}{2}\right)
$$
for every real number $x$ and every positive integer $n$.