Question

If the $pth,qth,rth$ and $sth$ terms of an A.P. are in G.P. then $p-q,q-r,r-s$ are in .........

• A. A.P
• B. H.P
• C. G.P
• D. A.G.P

Answer 1

Answer: (C)

G.P

$pth,qth,rth$ and $sth$ terms of an A.P. are in G.P.

Let $a$ be first term and $d$ be common difference of A.P. and $R$ be common ratio of G.P.

$\Rightarrow a+(p-1)d,a+(q-1)d,a+(r-1)d,a+(s-1)d$ are in G.P.

$\Rightarrow R=\frac{a+(q-1)d}{a+(p-1)d}=\frac{a+(r-1)d}{a+(q-1)d}=\frac{a+(s-1)d}{a+(r-1)d}$

Consider $\frac{a+(q-1)d}{a+(p-1)d}=\frac{a+(r-1)d}{a+(q-1)d}$
Subtracting $1$ on both sides gives

$\Rightarrow \frac{a+(q-1)d}{a+(p-1)d}-1=\frac{a+(r-1)d}{a+(q-1)d}-1$

$\Rightarrow \frac{(q-p)d}{a+(p-1)d}=\frac{(r-q)d}{a+(q-1)d}$

$\Rightarrow R=\frac{q-r}{p-q}$ -----(1)

Similarly,
$R=\frac{r-s}{q-r}$ -----(2)

From (1) and (2)

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

$\therefore p-q,q-r,r-s$ are in G.P.

Hence, option C.