Value of i n - i n -1- i n -2- i n -3- when n - I is equal toA- 1B- -1 IC- 0D- 2

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Standard XII

Mathematics

Complex Numbers

Value of i n ...

Question

Value of $i^{n} + i^{n + 1} + i^{n + 2} + i^{n + 3}$ , when $n \in I$ is equal to

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Solution

The correct option is C $0$
$i^{n} + i^{n + 1} + i^{n + 2} + i^{n + 3}$
$= i^{n} + i^{n} . i + i^{n} . i^{2} + i^{n} . i^{3}$
$= i^{n} + i^{n} . i - i^{n} - i . i^{n} = 0$

$Alternate Solution$
As $n, n + 1, n + 2, n + 3$ are four consecutive integers.
$∵$ Sum of any four consecutive integral powers of $i$ is zero.
So, $i^{n} + i^{n + 1} + i^{n + 2} + i^{n + 3} = 0$

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Value of in+in+1+in+2+in+3, when n∈I is equal to

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