In class recently an identity was mentioned, and I am unsure of how they got the solution. The equation is the following:
$$\sum_{k=0}^{n}{2^{2k}} \binom{n}{k} = 5^n$$
I know I have to use the binomial coefficients but the results I have been getting just don't seem correct. Any idea how this works out? Thank you.
Got it! In the binomial theorem:
$$ (x + y)^n = \sum_{j=0}^n \binom{n}{j} x^j y^{n-j} $$
set $x = 4 = 2^2$ and $y = 1$. Done!