Question

If $(p+q{)}^{\text{th}}$ term and $(p-q{)}^{\text{th}}$ term of a G.P. be $m$ and $n,$ then the $p^{\text{th}}$ term will be ___.

• A. $\frac{m}{n}$
• B. $\sqrt{mn}$
• C. mn
• D. o

Answer 1

The correct option is B

$\sqrt{mn}$

Given that
$a_{p+q}=a{r}^{p+q-1}=m$ and $a_{p-q}=a{r}^{p-q-1}=n.$

$\Rightarrow m\times n=a{r}^{p+q-1}$ x $a{r}^{p-q-1}$

$=a^2{r}^{2(p-1)}$

$=(a{r}^{p-1}{)}^2$

$\Rightarrow \sqrt{mn}=a{r}^{p-1}=a_p$

Thus, the $p^{\text{th}}$ term of the GP is $\sqrt{mn}.$

Aliter: Each term in a G.P. is the geometric mean of the terms equidistant from it. Here, $(p+q{)}^{th}$ and $(p-q{)}^{th}$ terms are equidistant from the $p^{th}$ term.
$\therefore$ $p^{th}$ term ($\sqrt{mn}$) will be G.M. of $(p+q{)}^{th}$ and $(p-q{)}^{th}$ terms.