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The value of s u msum(n=1)^(13)(i^n+i^(...

The value of `s u msum_(n=1)^(13)(i^n+i^(n+1)),"w h e r e"i=sqrt(-1)"e q u a l s"` `i` (b) `i-1` (c) `-i` (d) `0`

A

i

B

i-1

C

`-i`

D

0

Text Solution

Verified by Experts

The correct Answer is:
D
`underset(n=1)overset(13)Sigma(i^n+i^(n+1))=underset(n=1)overset(13)Sigmai^n(1+i)=(1+i)underset(n=1)overset(13)Sigma i^n`
`=(1+i)(i+i^2+i^3.....+i^(13))=(1+i)[(i-(1-i^(13)))/(1-i)] `
`=(l+i)[(i(1-i))/(1-i)]=(1+i)=i-1`
Alternate Solution
Since sum of any four comsecutive powers fo iota is zero
` therefore underset(n=1) overset(13)Sigma (i^n+i^(n+1)+....+i^(13))`
`+(i^2+i^3 +.....+i^(14))=i+i^2=i-1`
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