Prove that


Prove that $\left(1+i^{10}+i^{20}+i^{30}\right)$ is a real number.



L.H.S $=\left(1+\mathrm{i}^{10}+\mathrm{i}^{20}+\mathrm{j}^{30}\right)$

$=\left(1+i^{4 \times 2+2}+i^{4 \times 5}+i^{4 \times 7+2}\right)$

Since $\Rightarrow i^{4 n}=1$

$\Rightarrow i^{4 n+1}=i$

$\Rightarrow i^{4 n+2}=-1$

$\Rightarrow i^{4 n+3}=-1$



= 0, which is a real no

Hence, $\left(1+\mathrm{i}^{10}+\mathrm{i}^{20}+\mathrm{i}^{30}\right)$ is a real number.


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