Question

If the pth, qth and rth terms of an A.P. are in G.P., then the common ratio of the G.P. is

• A. $\frac{p+q}{r+q}$
• B. $\frac{p-r}{p-q}$
• C. $\frac{r-q}{q-p}$
• D. none of these

Answer 1

Answer: (C)

$\frac{r-q}{q-p}$

Answer 2

Let $a$ be the first term and $d$ be the common difference of the $A.P$. $T_{p} = a+\left(p-1\right)d$ $T_{q} = a+\left(q-1\right)d$ $T_{r} = a+\left(r-1\right)d$ Since $T_{p} , T_{q}$ are in $G.P$. $\therefore\frac{T_{q}}{T_{p}} = \frac{T_{r}}{T_{q}} = R$ (Common ratio of $G.P$.) $\therefore\frac{ a+\left(q-1\right)d}{a+\left(p-1\right)d} =\frac{ a+\left(r-1\right)d}{a+\left(q-1\right)d} = \frac{R}{1}$ $\therefore \frac{a+\left(q-1\right)d}{\left(p-q\right)d} = \frac{R}{1-R}$ and $\frac{\left(r-q\right)d}{a+\left(q-1\right)d} = \frac{R-1}{1}$ $\therefore$ Multiply these two $\frac{r-q}{p-q} = -R$ $\therefore R =\frac{ r-q}{q-p}.$