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Question
Find the value of the following expression:
i5 + i10 + i15
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Solution
\[ i^5 + i^{10} + i^{15} \]
\[ = i^{4 \times 1 + 1} + i^{4 \times 2 + 2} + i^{4 \times 3 + 3} \]
\[ = \left\{ \left( i^4 \right)^1 \times i \right\} + \left\{ \left( i^4 \right)^2 \times i^2 \right\} + \left\{ \left( i^4 \right)^3 \times i^3 \right\}\]
\[ = i + i^2 + i^3 \left[ \because i^4 = 1 \right]\]
\[ = i - 1 - i \left[ \because i^2 = - 1, i^3 = - i \right] \]
\[ = - 1\]
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