For positive integers n_1, n_2, the value of | vedclass

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(D) Given the expression: $(1 + i)^{n_1} + (1 + i^3)^{n_1} + (1 + i^5)^{n_2} + (1 + i^7)^{n_2}$.Since $i^3 = -i$, $i^5 = i$, and $i^7 = -i$, the expre

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$n_1 > 0, n_2 > 0$

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(D) Given the expression: $(1 + i)^{n_1} + (1 + i^3)^{n_1} + (1 + i^5)^{n_2} + (1 + i^7)^{n_2}$.Since $i^3 = -i$, $i^5 = i$, and $i^7 = -i$, the expression becomes:$(1 + i)^{n_1} + (1 - i)^{n_1} + (1 + i)^{n_2} + (1 - i)^{n_2}$.Let $z = (1 + i)^{n} + (1 - i)^{n}$.Using the polar form $1 + i = \sqrt{2} e^{i\pi/4}$ and $1 - i = \sqrt{2} e^{-i\pi/4}$, we get:$z = (\sqrt{2})^n (e^{in\pi/4} + e^{-in\pi/4}) = 2^{n/2} \cdot 2 \cos(n\pi/4) = 2^{n/2+1} \cos(n\pi/4)$.Since $2^{n/2+1} \cos(n\pi/4)$ is always a real number for any positive integer $n$, the sum of two such terms is also always real.Therefore, the expression is a real number for all positive integers $n_1$ and $n_2$.

List-$I$	List-$II$
$(i)$ $a \bar{a}$	$(A)$ $-\frac{\pi}{3}$
$(ii)$ $\arg \left(\frac{1}{\bar{a}}\right)$	$(B)$ $-i \sqrt{3}$
$(iii)$ $a - \bar{a}$	$(C)$ $2i / \sqrt{3}$
$(iv)$ $\operatorname{Im}\left(\frac{4}{3a}\right)$	$(D)$ $1$
	$(E)$ $\pi / 3$
	$(F)$ $\frac{2}{\sqrt{3}}$

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 and only if:

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