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Question
Show that 1 + i10 + i20 + i30 is a real number.
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Solution
\[1 + i^{10} + i^{20} + i^{30} \]
\[ = 1 + i^{4 \times 2 + 2} + i^{4 \times 5} + i^{4 \times 7 + 2} \]
\[ = 1 + \left[ \left( i^4 \right)^2 \times i^2 \right] + \left( i^4 \right)^5 + \left[ \left( i^4 \right)^7 \times i^2 \right]\]
\[ = 1 + i^2 + 1 + i^2 \left( \because i^4 = 1 \right)\]
\[ = 1 - 1 + 1 - 1 \left( \because i^2 = - 1 \right)\]
\[ = 0\]
\[\text { This is a real number} . \]
\[\text { Hence proved } .\]
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