The value of the sum `sum_(n=1)^(13) (i^(n)+i^(n+1))where i =sqrt(-1)` is:

Options

A. `i`
B. `-i`
C. 0
D. `i-1`

Answer

D

Explanation

`underset(n=1)overset(13)(sum)[i^(n)+i^(n+1)]=underset(n=1)overset(13)(sum)i^(n)[1+i]`
`=(1+i)[i+i^(2)+i^(3)....i^(13)]=((1+i))/((1-i))i[1-i^(13)]`
`=((-1+i)(1-i^(13)))/((1-i))=(-1i^(13)+i-i^(14))/((1-i))`
`=(-1+(i^(2))^(6)i+11-(i^(2))^(7))/((1-i))=(2i+2i^(2))/(1-i^(2))=(1-i)`