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Question
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
Solution
\[(vi) 1 + i^2 + i^4 + i^6 + i^8 + . . . + i^{20} \]
\[ \because i^2 = - 1, \]
\[ i^4 = 1, \]
\[ i^6 = - 1, \]
\[ i^8 = 1, \]
\[ i^{20} = 1\]
\[ \therefore 1 + i^2 + i^4 + i^6 + i^8 + . . . + i^{20} \]
\[ = \left[ 1 + \left( - 1 \right) \right] + \left[ 1 + \left( - 1 \right) \right] + \left[ 1 + \left( - 1 \right) \right] + . . . + \left[ 1 + \left( - 1 \right) \right] + 1\]
\[ = 5 \times \left[ 1 + \left( - 1 \right) \right] + 1 \left[ \text { As, there are 11 terms} \right]\]
\[ = 5 \times 0 + 1\]
\[ = 1\]
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