Value of sum Σlimits_{n = 1}^{13} {({i^n} + {i^{n + 1}})} , where i = √ {

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4-1.Complex numbers

medium

The value of the sum $\sum\limits_{n = 1}^{13} {({i^n} + {i^{n + 1}})} $, where $i = \sqrt { - 1} $, equals

$i$

$i - 1$

$ - i$

$0$

(IIT-1998)

Solution

(b) $\sum\limits_{n = 1}^{13} {({i^n} + {i^{n + 1}})} $$ = (i + {i^2} + {i^3} + …. + {i^{13}}) + ({i^2} + {i^3} + …. + {i^{14}})$
$ = \frac{{i(1 – {i^{13}})}}{{1 – i}} + \frac{{{i^2}(1 – {i^{13}})}}{{1 – i}}$$ = i\,\left( {\frac{{1 – i}}{{1 – i}}} \right) + \frac{{{i^2}(1 – i)}}{{(1 – i)}}$
$ = i + {i^2} = i – 1$.

Standard 11

Mathematics

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The value of the sum $\sum\limits_{n = 1}^{13} {({i^n} + {i^{n + 1}})} $, where $i = \sqrt { - 1} $, equals

Solution

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