Question

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

• A. $n_{1}=n_{2}+1$
• B. $n_{1}=n_{2}-1$
• C. $n_{1}=n_{2}$
• D. $n_{1}>0, n_{2}>0$

Answer 1

Answer: (D)

$(1+i)^{n_{1}}+\left(1+i^{3}\right)^{n_{1}}+\left(1+i^{5}\right)^{n_{2}}+\left(1+i^{7}\right)^{n_{2}}$
$=(1+i)^{n_{1}}+(1-i)^{n_{1}}+(1+i)^{n_{2}}+(1-i)^{n_{2}}$
$=2^{n_{1} / 2}(\cos \pi / 4+i \sin \pi / 4)^{n_{1}}+2^{n_{1} / 2}(\cos \pi / 4-i \sin \pi / 4)^{n_{1}}$
$+2^{n_{2} / 2}(\cos \pi /+i \sin \pi / 4)^{n_{2}}+2^{n_{2} / 2}(\cos \pi / 4-i \sin \pi / 4)^{n_{2}}$
${[\because 1 \pm i=\sqrt{2}(\cos \pi / 4 \pm i \sin \pi / 4)] }$
$=2^{n_{1} / 2}\left(\cos \frac{n_{1} \pi}{4}+i \sin \frac{n_{1} \pi}{4}\right)+2^{n_{1} / 2}\left(\cos \frac{n_{1} \pi}{4}-i \sin \frac{n_{1} \pi}{4}\right)$
$+2^{n_{2} / 2}\left(\cos \frac{n_{2} \pi}{4}+i \sin \frac{n_{2} \pi}{4}\right)+2^{n_{2} / 2}\left(\cos \frac{n_{2} \pi}{4}-i \sin \frac{n_{2} \pi}{4}\right)$
$=2 \cdot 2^{n_{1} / 2}\left(\cos \frac{n_{1} \pi}{4}\right)+2 \cdot 2^{n_{2} / 2} \cos \frac{n_{2} \pi}{4}$
which is real for all $n_{1}>0, n_{2}>0$