A) \[i\]
B) \[-i\]
C) \[0\]
D) \[i-1\]
Correct Answer: D
Solution :
\[\sum\limits_{n=1}^{13}{\left| {{i}^{n}}+{{1}^{n+1}} \right|}=\sum\limits_{n=1}^{13}{{{i}^{n}}\left[ 1+i \right]}\] \[=\,\,\,(1+i)\left[ i+{{i}^{2}}+{{i}^{3}}....{{i}^{13}} \right]=\frac{(1+i)}{(1-i)}i\left[ 1-{{i}^{13}} \right]\]\[=\,\,\,\frac{(-1+i)(1-{{i}^{13}})}{(1-i)}=\frac{-1+{{i}^{13}}+i-{{i}^{14}}}{(1-i)}\] \[=\,\,\,\frac{-1+{{({{i}^{2}})}^{6}}.i+i-{{({{i}^{2}})}^{7}}}{(1-i)}=\,\,\frac{2i+2{{i}^{2}}}{1-{{i}^{2}}}=(i-1)\]
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