Complex Numbers And Quadratic Equations question 396
Question: The value of the sum $ \sum\limits_{n=1}^{13}{(i^{n}+{i^{n+1}})} $ , where $ i=\sqrt{-1} $ , equals [IIT 1998]
Options:
A) $ i $
B) $ i-1 $
C) $ -i $
D) 0
Show Answer
Answer:
Correct Answer: B
Solution:
$ \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( \frac{1-i}{1-i} )+\frac{i^{2}(1-i)}{(1-i)} $ $ =i+i^{2}=i-1 $ .
BETA
