Question

$1,5,25$ are the pth , qth and rth terms respectively of a G.P Prove that $p,q,r$ in -

• A. A.P.
• B. G.P.
• C. H.P.
• D. A.G.P.

Answer 1

The correct option is A A.P.

Let first term of GP$=$a

and common ratio $=$b

then as given

$1=a{b}^{b-1}\to \left(1\right)$

$5=a{b}^{9-1}\to \left(2\right)$

$25=a{b}^{r-1}\to \left(3\right)$

Take log on both sides of $\left(1\right)$, we get

$log1=loga+(p-1)logb$

$\therefore p=\frac{-loga}{logb}+1$

Similarly take log on $\left(2\right)$, we get

$log5=loga+(q-1)logb$

$\therefore q=\frac{log5-loga}{logb}+1$

and $r=\frac{log25-loga}{logb}+1\frac{2log5-loga}{logb}+1$

Clearly $r-q=\frac{log5}{logb}$

and $q-p=\frac{log5}{logb}$

as $r-q=q-p$

$\therefore$ p, q, r are in AP.