Value of sum $\sum_{i=0}^n\binom{2n}{2i}(-3)^i$

Question

What is the value of the sum $\sum_{i=0}^n\binom{2n}{2i}(-3)^i$?

It looks like the binomial expansion $(1+x)^n=\sum_{i=0}^n\binom{n}{i}x^i$, but we only take every other term, and also the power is $(-3)^i$, not $(-3)^{2i}$.

Answer 1

You're almost there. You can use these two functions:

$$\frac{f(x) + f(-x)}{2} \tag{1}$$

and

$$(\sqrt{-3})^{2i} = (-3)^i \tag{2}$$

Can you see what (1) does to the terms? If you can't, you can try writing out the terms like so:

$$f(x) = c_0 x^0 + c_1 x^1 + c_2 x^2 + \dots$$

Then find what $f(-x)$ gives you.