Question:
Find the sum $\left(\mathrm{i}^{\mathrm{n}}+\mathrm{i}^{\mathrm{n}+1}+\mathrm{i}^{\mathrm{n}+2}+\mathrm{i}^{\mathrm{n}+3}\right)$, where $\mathrm{n} \mathrm{N}$.
Solution:
We have $\mathrm{i}^{\mathrm{n}}+\mathrm{i}^{\mathrm{n}+1}+\mathrm{i}^{\mathrm{n}+2}+\mathrm{i}^{\mathrm{n}+3}$
$=i^{n}+i^{n} \cdot i+i^{n} \cdot i^{2}+i^{n} \cdot i^{3}$
$=i^{n}\left(1+i+i^{2}+i^{3}\right)$
$=i^{n}(1+i-1-i)$
$=i^{n}(0)=0$
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