$\sum_{k=0}^{n} \binom{n}{2k}3^{n-2k}2^{3k}$

Asked Mar 24 '22 at 13:31

Active Mar 24 '22 at 13:31

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I was solving a question on matrices and ended up on this sum $$ 2\sum_{k=0}^{n} \binom{n}{2k}3^{n-2k}2^{3k} $$

I tried opening the summation and writing the terms to find any hints, but couldn't find any.

The answer given is $(3+2 \sqrt{2})^n+(3-2 \sqrt{2})^n$

Any hint would suffice considering I'm a high school student and don't know a lot about higher math.

asked Mar 24 '22 at 13:31

Melo

Hint: Expand $(a+b)^n + (a-b)^n$. – Calvin Lin Mar 24 '22 at 13:33
You might want to take a look at series multisection and roots of unity filter if you want to learn more about problems like this. – Joseph Camacho Mar 24 '22 at 13:37

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