Question

The Geometric mean of first $2n$ term if the series $a,ar,a{r}^2\cdots$ is given by

• A. $a{r}^{\frac{2n-1}{2}}$
• B. $a{r}^{\frac{n-1}{2}}$
• C. $a^{\frac{n}{2}}$
• D. $a{r}^{n-1}$

Answer 1

The correct option is A $a{r}^{\frac{2n-1}{2}}$

Required mean $={(a.ar.a{r}^2.a{r}^3.............a{r}^{2n-1})}^{1/2n}$
$={\left(a^{2n}{r}^{1+2+3+...........+(2n-1)}\right)}^{1/2n}=a{r}^{\frac{\frac{2n-1}{2}(1+(2n-1\left).1\right)}{2n}}=a{r}^{\frac{2n-1}{2}}$