A) 2
B) \[-2\]
C) 0
D) 1
E) \[-1\]
Correct Answer: C
Solution :
AM of a and b is\[\frac{a+b}{2}\]. \[\Rightarrow \] \[\frac{{{a}^{n+1}}+{{b}^{n+1}}}{{{a}^{n}}+{{b}^{n}}}=\frac{a+b}{2}\] \[\Rightarrow \] \[2({{a}^{n+1}}+{{b}^{n+1}})=({{a}^{n}}+{{b}^{n}})(a+b)\] \[\Rightarrow \] \[2{{a}^{n+1}}+2{{b}^{n+1}}={{a}^{n+1}}+{{a}^{n}}b\]\[+{{b}^{n}}a+{{b}^{n+1}}\] \[\Rightarrow \] \[{{a}^{n+1}}+{{b}^{n+1}}={{a}^{n}}b+{{b}^{n}}a\] \[\Rightarrow \] \[{{a}^{n}}(a-b)={{b}^{n}}(a-b)\] \[\Rightarrow \] \[{{a}^{n}}={{b}^{n}}\] \[\Rightarrow \] \[\frac{{{a}^{n}}}{{{b}^{n}}}=1\Rightarrow {{\left( \frac{a}{b} \right)}^{n}}={{\left( \frac{a}{b} \right)}^{0}}\] \[\Rightarrow \] \[n=0\]
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