The value of `1+ sum_(k=0)^(14){cos frac((2k+1)pi)(15) + i sin frac ((2k + 1) pi)(15)}` is

Options

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

Answer

C

Explanation

We have,
`1+sum_(k=0)^(14) {cos((2k+1)pi)/(15)-I sin((2k+1)pi)/(15)}`
`=1+sum_(k=0)^(14) e^(i(2k+1)pi)/15`
`1+(alpha+alpha^(3)+alpha^(5)+……….+alpha^(29))`
`=1+alpha{((1-alpha^(2))^(15))/(1-alpha^(2))}`
`=1+alpha(1-alpha^(30))/(1-alpha^(2))=1+alpha(1-1)/(1-alpha^(2))=1` `[therefore alpha^(30)=e^(i2pi)=1]`