Question

If mth, nth and pth terms of a G.P. form three consecutive terms of a G.P. Then, m, n and p form

• A. $G.P$
• B. $H.P$
• C. $A.P$
• D. None of these

Answer 1

The correct option is C $A.P$

Let $a$ be the first term and $r$ be the common ratio of the G.P. Then,

$a_m=a{r}^{m-1},a_n=a{r}^{n-1},a_p=a{r}^{p-1}$

It is given that $a_m,a_n,a_p$ are in GP.

Therefore,

$(a_n{)}^2=a_m{a}_p$

$(a{r}^{n-1}{)}^2=(a{r}^{m-1}\times a{r}^{p-1})$

$a^2{r}^{2n-2}=a^2{r}^{m+p-2}$

$r^{2n-2}=r^{m+p-2}$

$2n-2=m+p-2$

$2n=m+p$

$m,n,p$ are in A.P.