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Question
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
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Solution
\[ i^{49} + i^{68} + i^{89} + i^{110} = i^{4 \times 12 + 1} + i^{4 \times 17} + i^{4 \times 22 + 1} + i^{4 \times 27 + 2} \]
\[ = \left[ \left( i^4 \right)^{12} \times i \right] + \left[ \left( i^4 \right)^{17} \right] + \left[ \left( i^4 \right)^{22} \times i \right] + \left[ \left( i^4 \right)^{27} \times i^2 \right]\]
\[ = i + 1 + i - 1 \left( \because i^4 = 1, i^2 = - 1 \right)\]
\[ = 2i\]
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