Question:
If $n \in \mathbb{N}$, then find the value of $i^{n}+i^{n+1}+i^{n+2}+i^{n+3}$.
Solution:
$i^{n}+i^{n+1}+i^{n+2}+i^{n+3}$
$=i^{n}+i^{n} \cdot i+i^{n} \cdot i^{2}+i^{n} \cdot i^{3}$
$=i^{n}+i^{n} \cdot i+i^{n} \cdot(-1)+i^{n} \cdot(-i)$
$=i^{n}+i^{n} \cdot i-i^{n}-i^{n} \cdot i$
$=0$
Thus, the value of $i^{n}+i^{n+1}+i^{n+2}+i^{n+3}$ is 0 .
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