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Question
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
Options
\[\frac{1}{2}(1 + i)\]
\[\frac{1}{2}(1 - i)\]
1
\[\frac{1}{2}\]
Solution
\[\frac{1}{2}(1 + i)\]
\[\frac{i^5 + i^6 + i^7 + i^8 + i^9}{1 + i}\]
\[ = \frac{i - 1 - i + 1 + i}{1 + i}\left[ \text { As,} i^5 = i, i^6 = - 1, i^7 = - i, i^8 = 1, i^9 = i \right]$\]
\[ = \frac{i}{i + 1}\]
\[ = \frac{i}{i + 1} \times \frac{i - 1}{i - 1}\]
\[ = \frac{i\left( i - 1 \right)}{i^2 - 1}\]
\[ = \frac{i^2 - i}{- 2}\]
\[ = \frac{1}{2}\left( 1 + i \right)\]
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