Let z be a non - real complex number which satisfies the equation ` z^(23) = 1 `. Then the value of ` sum_(22)^(k = 1 ) (1)/(1 + z ^( 8k) + z ^( 16k )) `

Answer

15

Explanation

`sum_(k=1)^(22) (1)/(1+z^(8k) +z^(16k)) = sum_(k=1)^(22) (z^(8k)-1)/((z^(8k) -1)(z^(16) + z^(8k) +1))`
`= sum_(k=1)^(22)(z^(8k)-1)/(z^(24k)-1)`
`=sum_(k=1)^(22) (z^(8k)-1)/(z^(k)-1)" "(because z^(24k)=z^(23k) z^(k) = 1.z^(k) = z^(k))`
`=sum_(k=1)^(22)(1+z^(k) + z^(2k) +......+z^(7k))` ltbvrgt `=22 +(0-1)xx7`
= 15