Question

If `i^2=-1` then the sum of `cos45+icos135+…i^n*cos(45+90n)+…i^40*cos3645=`?

Answer 1

`cos45+icos135+…i^n*cos(45+90n)+…i^40*cos3645=?`

Sum of First 4 terms

`sum_(r=1)^(r=4)S_r`

`=cos45+icos135+i^2cos(45+90xx2)+i^3cos(45+90xx3)`

`=cos45-icos45+cos45-icos45`

`=1/sqrt2-i/sqrt2+1/sqrt2-i/sqrt2=sqrt2(1-i)`

Sum of Second 4 terms

`sum_(r=5)^(r=9)S_r`

`=i^4cos(45+90xx4)+i^5cos(45+90xx5)+i^6cos(45+90xx6)+i^7cos(45+90xx7)`

`=cos45-icos45+cos45-icos45`

`=1/sqrt2-i/sqrt2+1/sqrt2-i/sqrt2=sqrt2(1-i)`

Considering this way we get 10 groups of 4terms having same sum.

So our sum

`S=sum_(r=1)^(r=40)S_r+i^40*cos3645`

`=10xxsqrt2(1-i)+cos(45+90xx40)`

`=10sqrt2(1-i)+cos45`

`=10sqrt2(1-i)+1/sqrt2`