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The value of $\sum\limits_{n=1}^{13} (i^n+i^{n+1})=?$

$\begin{array}{1 1}(A)i \\ (B) -i \\ (C) 0 \\(D) i-1 \end{array}$

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  • $x+x^2+x^3+.............x^n=\large\frac{x(1-x^n)}{1-x}$
$\sum\limits_{n=1}^{13} (i^n+i^{n+1})=\sum\limits_{n=1}^{13} i^n(1+i)$
$=(1+i)(i+i^2+i^3+...............i^{13})$
$=(1+i)\large\frac{i(1-i^{13})}{1-i}$
$=(1+i)i$ (since $i^{13}=i$)
$=i-1$
answered Jul 28, 2013 by rvidyagovindarajan_1
 

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