Question

The expression $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}is[a\ne b\ne 0]$ is (where a and b are unequal non-zero numbers)

• A. A.M. between a and b if n = -1
• B. G.M. between a and b$=-\frac{1}{2}$
• C. H.M. between a and b if n = 0
• D. all are correct

Answer 1

The correct option is B G.M. between a and b$=-\frac{1}{2}$

$Let\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\frac{a+b}{2}\Rightarrow 2a^{n+1}+2b^{n+1}=a^{n+1}+b^{n+1}+a{b}^n+b{a}^n\Rightarrow a^{n+1}-a^{n}b+b^{n+1}-a{b}^n=0\Rightarrow (a-b)(a^n-b^n)=0\Rightarrow a^n=b^n,\text{it is possible for unequal numbers a and b if n = 0}Let\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\sqrt{ab}\Rightarrow a^{n+1}+b^{n+1}=a^{n+\frac{1}{2}}{b}^{\frac{1}{2}}+a^{\frac{1}{2}}{b}^{n+\frac{1}{2}}(a^{n+\frac{1}{2}}-b^{n+\frac{1}{2}})(\sqrt{a}-\sqrt{b})=0\Rightarrow a^{n+\frac{1}{2}}-b^{n+\frac{1}{2}}=0,\text{which holds true if}n+\frac{1}{2}=0\Rightarrow n=-\frac{1}{2}Let\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\frac{2ab}{a+b}\Rightarrow a^{n+2}+a^{n+1}b+a{b}^{n+1}+b^{n+2}=2a^{n+1}b+2a{b}^{n+1}\Rightarrow (a-b)(a^{n+1}-b^{n+1})=0\Rightarrow a^{n+1}-b^{n+1}=0\Rightarrow n=-1$