For positive integers n₁,n₂ the value of the expression

is a real number if and only if ( $1 + i)^{n_{1}}$ +( $1 + i^{3})^{n_{1}}$ +( $1 + i^{5})^{n_{2}}$ ( $1 + i^{7})^{n_{2}}$ where i= $\sqrt[2]{- 1}$ is a real number if and only if

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Solution

The correct option is D

Using $i^{3}$ =-i, $i^{5}$ =i and $i^{7}$ =-i, we can write the

given expression as

( $1 + i)^{n_{1}}$ +( $1 + i^{3})^{n_{1}}$ +( $1 + i^{5})^{n_{2}}$ ( $1 + i^{7})^{n_{2}}$ where i= $\sqrt[2]{- 1}$

=2[ $c_{0}^{n_{1}}$ + $c_{1}^{n_{1}}$ $i^{2}$ + $c_{1}^{n_{1}}$ $i^{4}$ +....]+2[ $c_{0}^{n_{2}}$ + $c_{1}^{n_{2}}$ $i^{2}$ + $c_{1}^{n_{2}}$ $i^{4}$ +....]

=2[ $c_{0}^{n_{1}}$ - $c_{1}^{n_{1}}$ + $c_{1}^{n_{1}}$ +....]+2[ $c_{0}^{n_{2}}$ - $c_{1}^{n_{2}}$ + $c_{1}^{n_{2}}$ +....]

This is a real number irrespective of values of $n_{1}$ and $n_{2}$ .

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