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# If a ^ n + b^n/ a ^ n - 1+b^ n- 1 is the A.M. between a and b, then find the value of n.

15.   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.

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Given :  $\frac{a^n + b ^n }{a ^{ n-1}+ b ^{n-1}}$  is the A.M. between a and b.

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

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

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

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

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

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

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

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

$\Rightarrow \,\left [ \frac{a}{b} \right ]^{n-1}= 1$

$\Rightarrow \,n-1=0$

$\Rightarrow \,n=1$

Thus, value of n is 1.

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