Question

If $p^{th}$,$q^{th}$,$r^{th}$ and $s^{th}$ terms of an A.P. be in G.P., then (p - q),(q - r),(r - s) will be in

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

Answer 1

The correct option is A

G.P

If a and d be the first term and common difference of the A.P.

Then $T_p$ = a + (p - 1)d, $T_q$ = a + (q - 1)d and

$T_r$ = a+(r - 1)d.

If $T_p,T_q,T_r$ are in G.P.

Then its common ratio R = $\frac{T_q}{T_p}$ = $\frac{T_r}{T_q}$ = $\frac{T_q-T_r}{T_p-T_q}$

= $\frac{[a+(q-1\left)d\right]-[a+(r-1\left)d\right]}{[a+(p-1\left)d\right]-[a+(q-1\left)d\right]}$ = $\frac{q-r}{p-q}$

Similarly, we can show that R =$\frac{q-r}{p-q}$ = $\frac{r-s}{q-r}$

Hence (p - q), (q - r),(r - s) be in G.P.