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Question
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
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Solution
\[ i^{30} + i^{40} + i^{60} = i^{4 \times 7 + 2} + i^{4 \times 10} + i^{4 \times 15} \]
\[ = \left[ \left( i^4 \right)^7 \times i^2 \right] + \left[ \left( i^4 \right)^{10} \right] + \left[ \left( i^4 \right)^{15} \right]\]
\[ = - 1 + 1 + 1 \left( \because i^4 = 1, i^2 = - 1 \right)\]
\[ = 1 \]
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