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 -1 is a real number if and only ifA- n1-n2-1B- n1-n2-1- n1-n2D- n1-0- n2-0

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Binomial Theorem

For positive ...

Question

For positive integers n₁,n₂ 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= $\sqrt[2]{- 1}$ is a real number if and only if

=+1

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=-1

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>0,>0

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Solution

The correct option is D
>0,>0

Using $i^{3} = - i, i^{5} = i a n d i^{7} = - i$ , we can write the given expression as
$(1 + i)^{n_{1}} + (1 + i^{3})^{n_{1}} + (1 + i^{5})^{n_{2}} + (1 + i^{7})^{n_{2}} w h e r e i = \sqrt[2]{1} = 2 [^{n_{1}} C_{0} +^{n_{1}} C_{2} (i)^{2} +^{n_{1}} C_{4} (i)^{4} +^{n_{1}} C_{6} (i)^{6} + \dots \dots] + 2 [^{n_{2}} C_{0} +^{n_{2}} C_{2} (i)^{2} +^{n_{2}} C_{4} (i)^{4} +^{n_{2}} C_{6} (i)^{6} + \dots \dots] = 2 [^{n_{1}} C_{0} -^{n_{1}} C_{2} +^{n_{1}} C_{4} -^{n_{1}} C_{6} + \dots \dots] + 2 [^{n_{2}} C_{0} -^{n_{2}} C_{2} +^{n_{2}} C_{4} -^{n_{2}} C_{6} + \dots \dots]$
This is a real number irrespective of values of $n_{1}$ and $n_{2}$ .

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