Question

In a GP of n terms, if n is odd then the product of all terms = $M^n$ , where M is the middle term.

Answer 1

G.P. of n terms,n is odd

Let n $=2k+1$

Product of all terms $=a.ar.a{r}^2....a{r}^{n-1}$

$=a^n{r}^{\frac{n(n-1)}{2}}=(a{r}^{\frac{n-1}{2}}{)}^n=(a{r}^{\left(\frac{n+1}{2}\right)-1}{)}^n$

As n is odd,Middle term of G.P. $=a{r}^{\frac{n+1}{2}-1}$

Product of all terms $=(Middleterm{)}^n$