For positive integers $n_{1}, n_{2}$ the value of the expression $(1 + i)^{n_{1}} + (1 + i^{3})^{n_{1}} + (1 + i^{5})^{n_{2}} + (1 + i^{7})^{n_{2}}$ , where $i = \sqrt{- 1}$ , is a real number if

n_{1} = n_{2} + 1

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n_{1} = n_{2} - 1

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n_{1} = n_{2}

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n_{1} > 0, n_{2} > 0

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Solution

The correct option is D $n_{1} > 0, n_{2} > 0$
$(1 + i)^{n_{1}} + (1 + i^{2})^{n_{2}} + (1 + i^{5})^{n_{2}} + (1 + i^{7})^{n_{2}}$
$= (1 + i)^{n_{1}} + (1 - i)^{n_{2}} + (1 + i)^{n_{2}} + (1 - i)^{n_{2}}$
$= 2 [1 +^{n_{1}} C_{2} i^{2} +^{n_{1}} C_{4} i^{4} . . .] + 2 [1 +^{n_{2}} C_{2} i^{2} +^{n_{2}} C_{4} i^{4} . . .]$
$= 2 [1 + -^{n_{1}} C_{2} +^{n_{1}} C_{4} - . . .] + 2 [1 -^{n_{2}} C_{2} +^{n_{2}} C_{4} - . . .]$
Hence
For all $n_{1} > 0$ and $n_{2} > 0$ the above expression yields real integral number.
Where $n_{1}, n_{2} ϵ N$ .
Hence, option 'D' is correct.

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