Sequence And Series Question 257
Question: If the geometric mean between $ a $ and $ b $ is $ \frac{{a^{n+1}}+{b^{n+1}}}{a^{n}+b^{n}} $ , then the value of n is
Options:
A) 1
B) -1/2
C) 1/2
D) 2
Show Answer
Answer:
Correct Answer: B
Solution:
As given $ \frac{{a^{n+1}}+{b^{n+1}}}{a^{n}+b^{n}}={{(ab)}^{1/2}} $
$ \Rightarrow $ $ {a^{n+1}}-{a^{n+1/2}}{b^{1/2}}+{b^{n+1}}-{a^{1/2}}{b^{n+1/2}}=0 $
$ \Rightarrow $ $ ({a^{n+1/2}}-{b^{n+1/2}})({a^{1/2}}-{b^{1/2}})=0 $
$ \Rightarrow $ $ {a^{n+1/2}}-{b^{n+1/2}}=0 $ $ (\because \ a\ne b\Rightarrow {a^{1/2}}\ne {b^{1/2}}) $
$ \Rightarrow $ $ {{( \frac{a}{b} )}^{n+1/2}}=1={{( \frac{a}{b} )}^{0}}\Rightarrow n+\frac{1}{2}=0\Rightarrow n=-\frac{1}{2} $ .
BETA
