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Value of sum(k = 1)^(100)(i^(k!) + omega...

Value of `sum_(k = 1)^(100)(i^(k!) + omega^(k!))`, where i = `sqrt(-1) and omega` is complex cube root of unity , is :

A

`190+omega`

B

`192+omega^(2)`

C

`190+i`

D

`192+i`

Text Solution

Verified by Experts

The correct Answer is:
D
`sum_(k=1)^(100) i^(k!) + sum_(k=1)^(100) omega^(k!)`
` sum_(k=1)^(100) i^(k!) = i^(1!) + i^(2!) + i^(3!) + i^(4!) + ……. i^(100!)`
` =i-1 + i^(6) + 1+1 +1 + …….. +1`
`=i-2 + 97 = i + 95`.
` sum_(k=1)^(100) omega^(k!) = omega^(1!) + omega^(2!) + omega^(3!) + omega^(4!) + ......... omega^(100!)`
` = omega + omega^(2) + 1 + 1 +1 + .......... + 1 = 97`
sum = i + 95 + 97 = i + 192
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