Answer
$$i^{2017}+i^{2015}=i-i$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}i^{2017}+i^{2015}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}i-i\end{aligned} $$ | |
| ① | $$ i^{2017} = i^{4 \cdot 504 + 1} =
\left( i^4 \right)^{ 504 } \cdot i^1 =
1^{ 504 } \cdot i =
i $$ |
| ② | $$ i^{2015} = i^{4 \cdot 503 + 3} =
\left( i^4 \right)^{ 503 } \cdot i^3 =
1^{ 503 } \cdot (-i) =
-i = -i $$ |
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