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UPSC NDA I 2017 Mathematics Solved Paper

Question : 14 of 120

Marks: +1, -0

The value of

i^{2n} + i^{2n+1} + i^{2n+2} + i^{2n+3}

i=\sqrt{-1}

, is

Solution:

Concept:

The powers of

i

repeat in a cycle of 4:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

, and then cycle again.

Explanation:

Consider the given expression:

i^{2n} + i^{2n+1} + i^{2n+2} + i^{2n+3}

Factor out

i^{2n}

from all four terms:

i^{2n}(1 + i + i^2 + i^3)

Now simplify the sum inside the parentheses using the known values of powers of

i

1 + i + i^2 + i^3 = 1 + i + (-1) + (-i)

The real parts add:

1 + (-1) = 0

. The imaginary parts add:

i + (-i) = 0

Therefore the entire bracket equals

0

Multiplying by

i^{2n}

gives

i^{2n} \times 0 = 0

The result is independent of the integer

n

and always zero.

Answer:

0

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