Complex Numbers And Quadratic Equations question 707
Question: The value of the sum $ \sum\limits_{n=1}^{13}{( i^{n}+{i^{n+1}} )}; $ where $ i=\sqrt{-1} $ is:
Options:
A) $ i $
B) $ -i $
C) $ 0 $
D) $ i-1 $
Show Answer
Answer:
Correct Answer: D
Solution:
$ \sum\limits_{n=1}^{13}{| i^{n}+{1^{n+1}} |}=\sum\limits_{n=1}^{13}{i^{n}[ 1+i ]} $ $ =(1+i)[ i+i^{2}+i^{3}….i^{13} ]=\frac{(1+i)}{(1-i)}i[ 1-i^{13} ] $ $ =\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+2i^{2}}{1-i^{2}}=(i-1) $
BETA
