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Question
Find the value of the following expression:
i49 + i68 + i89 + i110
Solution
\[\ i^{49} + i^{68} + i^{89} + i^{110} \]
\[ = i^{4 \times 12 + 1} + i^{4 \times 17} + i^{4 \times 22 + 1} + i^{4 \times 27 + 2} \]
\[ = \left\{ \left( i^4 \right)^{12} \times i \right\} + \left\{ \left( i^4 \right)^{17} \right\} + \left\{ \left( i^4 \right)^{22} \times i \right\} + \left\{ \left( i^4 \right)^{27} \times i^2 \right\}\]
\[ = i + 1 + i + i^2 \left[ \because i^4 = 1 \right]\]
\[ = 2i + 1 - 1 \left[ \because i^2 = - 1 \right] \]
\[ = 2i\]
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