Question

If the geometric mean between a and b is $\frac{a^{n+1}+b^{n+1}}{a^n+b^n},$ then the value of n is

• A. 1
• B. –1/2
• C. 1/2
• D. 2

Answer 1

The correct option is B

–1/2

As given $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=(ab{)}^{1/2}$
$\Rightarrow (a^{n+1}-a^{n+1/2}{b}^{1/2}+b^{n+1}-a^{1/2}{b}^{n+1/2}=0$
$\Rightarrow (a^{n+1/2}-b^{n+1/2})(a^{1/2}-b^{1/2})=0$
$\Rightarrow (a^{n+1/2}-b^{n+1/2})=0(\because a\ne b\Rightarrow a^{1/2}\ne b^{1/2})$
$\Rightarrow {\left(\frac{a}{b}\right)}^{n+1/2}=1={\left(\frac{a}{b}\right)}^o\Rightarrow n+\frac{1}{2}=0\Rightarrow n=-\frac{1}{2}.$