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

n_{1} = n_{2}

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n_{2} = n_{2} - 1

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n_{1} = n_{2} + 1

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\forall n_{1}

and

n_{2}

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Solution

The correct option is D $\forall n_{1}$ and $n_{2}$
$(1 + i)^{n_{1}} =$ $^{n_{1}} C_{0} +^{n_{1}} C_{1} i +^{n_{1}} C_{2} i^{2} + . . . . . . +^{n_{1}} C_{n_{1}} i^{n_{1}}$ --------(1)

$(1 + i^{3})^{n_{1}} = (1 - i)^{n_{1}} =$ $^{n_{1}} C_{0} -^{n_{1}} C_{1} i +^{n_{1}} C_{2} i^{2} - . . . . . . +^{n_{1}} C_{n_{1}} i^{n_{1}}$ --------(2)

$(1 + i^{5})^{n_{2}} = (1 + i)^{n_{2}} =$ $^{n_{2}} C_{0} +^{n_{2}} C_{1} i +^{n_{2}} C_{2} i^{2} + . . . . . . +^{n_{2}} C_{n_{2}} i^{n_{2}}$ --------(3)

$(1 + i^{7})^{n_{2}} = (1 - i)^{n_{2}} =$ $^{n_{2}} C_{0} -^{n_{2}} C_{1} i +^{n_{2}} C_{2} i^{2} - . . . . . . +^{n_{2}} C_{n_{2}} i^{n_{2}}$ --------(4)

Adding (1),(2),(3) and (4),

$(1 + i)^{n_{1}} + (1 + i^{3})^{n_{1}} + (1 + i^{5})^{n_{2}} + (1 + i^{7})^{n_{2}}$
$= 2 (^{n_{1}} C_{0} +^{n_{1}} C_{2} i^{2} +^{n_{1}} C_{4} i^{4} + . . . . . . . . . . .) + 2 (^{n_{2}} C_{0} +^{n_{2}} C_{2} i^{2} +^{n_{2}} C_{4} i^{4} + . . . . . . . . . . .)$

$= 2 (^{n_{1}} C_{0} -^{n_{1}} C_{2} +^{n_{1}} C_{4} + . . . . . . . . . . .) + 2 (^{n_{2}} C_{0} -^{n_{2}} C_{2} +^{n_{2}} C_{4} + . . . . . . . . . . .)$
$\Rightarrow$ As there are only even powers of $i$ , the expression is real for all positive integers $n_{1}$ and $n_{2}$

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For positive integers n1 and n2 the value of the expression (1+i)n1+(1+i3)n1+(1+i5)n2+(1+i2)n2 where i=√−1 is a real number iff

For positive integers $n_{1}$ and $n_{2}$ the value of the expression $(1 + i)^{n_{1}} + (1 + i^{3})^{n_{1}} + (1 + i^{5})^{n_{2}} + (1 + i^{2})^{n_{2}}$ where
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