Question

In a GP of n terms, if n is even the product of all term = $M^n$ , where M is the geometric mean of the two middle terms.

Answer 1

$IfG.P.ofnterms,ifnisevenMiddleterms=a{r}^{\frac{n}{2}-1},a{y}^{\frac{n}{2}-2}productofallterms=a.ar.a{r}^2.....a{r}^{n-1}=a^n.r^{\frac{n(n-1)}{2}}=(a{r}^{\frac{n-1}{2}}{)}^n=\sqrt{{(a}^2{r}^{n-1}{)}^n}=(\sqrt{\left(a{r}^{\frac{n}{2}-1}\right)\left(a{r}^{\frac{n}{2}}\right)}{)}^n(G.M.oftwomiddleterms{)}^n$