Show that 1+i^ 10 +i^ 20 +i^ 30 is a real number. - Vidyadip

Question

Show that $1+\mathrm{i}^{10}+\mathrm{i}^{20}+\mathrm{i}^{30}$ is a real number.

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Answer

$\begin{aligned} & 1+i^{10}+i^{20}+i^{30} \\ & =1+\left(i^4\right)^2 \cdot i^2+\left(i^4\right)^5+\left(i^4\right)^7 \cdot i^2 \\ & =1+(1)^2(-1)+(1)^5+(1)^7(-1)\left[\because i^4=1, i^2=-1\right] \\ & =1-1+1-1 \\ & =0, \text { which is a real number. }\end{aligned}$

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