Find $\sum_{k=1}^{1010}(a_k^k+2+\frac{1}{a_k^k})^{2020}$

Question

Given a polynomial

$P(x)=x^{2020}+x^{2019}+x^{2018}+...+x^2+x+1$

with roots $a_1,a_2,a_3,...,a_{2020}$ Find the value of

$\sum_{k=1}^{1010}(a_k^k+2+\frac{1}{a_k^k})^{2020}$

we can write to be

$\sum_{k=1}^{1010}(a_k^{\frac{k}{2}}+\frac{1}{a_k^{\frac{k}{2}}})^{2020.2}$ = $\sum_{k=1}^{1010}(a_k^{\frac{k}{2}}+\frac{1}{a_k^{\frac{k}{2}}})^{4040}$ = $\sum_{k=1}^{1010}(a_k^k+1)^{4040}$ $(a_k^{-2020k})$

Please give me idea, i don't know where to start.

Answer 1

Hint to help you start: $$\left(b+\frac1b\right)^2 = b^2+2+\frac1{b^2}$$