Question

First term of a $G.P$ of $n$ terms is $a$ and the last term is l. The product of all the terms of $G.P$ is:

• A. $\frac{n}{2}(a+l)$
• B. $(a+l{)}^{n/2}$
• C. $(al{)}^n$
• D. $(al{)}^{n/2}$

Answer 1

The correct option is D $(al{)}^{n/2}$

$a,ar,.....a{r}^{n-1}$

$a{r}^{n-1}=l$

$r^{n-1}=\frac{l}{a}$

Product of all terms

$a\times ar+a{r}^2+...\dots ..a{r}^{n-1}$

$\Rightarrow a^n\times r^{1+2+.....n-1}$

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

$=a^n{\left(\frac{l}{a}\right)}^{\frac{n}{2}}=(al{)}^{\frac{n}{2}}$.