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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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