Prove that `underset(r=0)overset(n)sum""^(n)C_(r)(-1)^(r)[i^(r)+i^(2r)+i^(3r)+i^(4r)]`
`=2^(n) + 2^(n+1)cos (npi//4)` , where `i = sqrt(-1)`

Explanation

`underset(r=0)overset(n)sum.^(n)C_(r)(-1)^(r)[i^(r)+i^(2r)+i^(3r)+u^(4r)]`
`=underset(r=0)overset(n)sum.^(n)C_(r)(-1)^(r)[i^(r) + (-1)^(r) + (-i)^(r)+1]`
` = underset(r=0)overset(n)sum[.^(n)C_(r)(-1)^(r)+.^(n)C_(r)+.^(n)C_(r)i^(r)+.^(n)C_(r)(-1)^(r)]`
`= (1-r)^(n) + (1+1)^(n) + (1+i)^(n) + (1-i)^(n)`
`=(sqrt(2))^(n) (1/(sqrt(2))-i/(sqrt(2)))^(n) + (sqrt(2))^(n) (1/(sqrt(2))+(i)/(sqrt(2)))^(n) + 2^(n)`
`= (sqrt(2))^(n)(cos'pi/4-isin'(pi)/(4))^(n)+(sqrt(2))^(n)(cos'(pi)/(4) + isin'(pi)/(4))^(n)+2^(n)`
`=(sqrt(2))^(n)(cos'(npi)/(4) - isin'(npi)/(4))+(sqrt(2))^(n)(cos'(npi)/(4) + i sin'(npi)/(4)) + 2^(n)`
` = 2(sqrt(2))^(n) cos'(npi)/(4) + 2^(n)`.