If frac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}} is A.M. between a and b, then value of n .

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8. Sequences and Series

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If $\frac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}}$ is the $A.M.$ between $a$ and $b,$ then find the value of $n$.

A

$1$

B

$1$

C

$1$

D

$1$

Solution

$A.M.$ of $a$ and $b$ $=\frac{a+b}{2}$

According to the given condition,

$\frac{a+b}{2}=\frac{a^{n}+b^{n}}{a^{n-1}+b^{n-1}}$

$\Rightarrow(a+b)\left(a^{n-1}+b^{n-1}\right)=2\left(a^{n}+b^{n}\right)$

$\Rightarrow a^{n}+a b^{n-1}+b a^{n-1}+b^{n}=2 a^{n}+2 b^{n}$

$\Rightarrow a b^{n-1}+a^{n-1} b=a^{n}+b^{n}$

$\Rightarrow a b^{n-1}-b^{n}=a^{n}-a^{n-1} b$

$\Rightarrow b^{n-1}(a-b)=a^{n-1}(a-b)$

$\Rightarrow b^{n-1}=a^{n-1}$

$\Rightarrow\left(\frac{a}{b}\right)^{n-1}=1=\left(\frac{a}{b}\right)^{0}$

$\Rightarrow n-1=0$

$\Rightarrow n=1$

Standard 11

Mathematics

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