Sequence And Series Question 155
Question: Let $ n(>1) $ be a positive integer, then the largest integer $ m $ such that $ (n^{m}+1) $ divides $ (1+n+n^{2}+…….+n^{127}) $ , is
[IIT 1995]
Options:
A) 32
B) 63
C) 64
D) 127
Show Answer
Answer:
Correct Answer: C
Solution:
Since $ n^{m}+1 $ divides $ 1+n+n^{2}+…….+n^{127} $ Therefore $ \frac{1+n+n^{2}+……+n^{127}}{n^{m}+1} $ is an integer
$ \Rightarrow $ $ \frac{1-n^{128}}{1-n}\times \frac{1}{n^{m}+1} $ is an integer
$ \Rightarrow $ $ \frac{(1-n^{64})(1+n^{64})}{(1-n)(n^{m}+1)} $ is an integer when largest $ m=64 $ .
BETA
