Question

If the $p^{th},q^{th}$ & $r^{th}$ terms of an AP are in GP.Then the common ratio of the GP is

• A. $\frac{q-r}{p-q}$
• B. $\frac{q+r}{q-p}$
• C. $\frac{r-p}{p-q}$
• D. $\frac{q-r}{q-p}$

Answer 1

The correct option is A $\frac{q-r}{p-q}$

Let first term of A.P is $a$ and common difference is $d$

$\therefore T_p=a+(p-1)d{T}_q=a+(q-1)d{T}_r=a+(r-1)d$

are in G.P

Let the first of G.P be $x$ and common ratio be $y$
Then

$a+(p-1)d=x$ ...(1)

$a+(q-1)d=xy$ ...(2)

$a+(r-1)d=x{y}^2$

Subtract (1) from (2), we get

$(q-1)d-(p-1)d=xy-x$

$\Rightarrow (q-p)d=xy-x$ ...(4)

Subtract $\left(2\right)$ from $\left(3\right)$

$(r-1)d-(q-1)d={xy}^2-xy=y(xy-x)$

$\Rightarrow (r-q)d=y(xy-x)$ ...(5)

Divide $\left(4\right)$ by $\left(5\right),$ we get

$\frac{(q-r)}{(p-q)}=y$