Question

If the $$p^{th}, \ q^{th},$$ and $$r^{th}$$ terms of an A.P. are in G.P., then common ratio of the G.P. is

A
$$\dfrac{pr}{q^2}$$
B
$$\dfrac{r}{p}$$
C
$$\dfrac{q+r}{p+q}$$
D
$$\dfrac{q-r}{p-q}$$
Solution
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Correct option is D. $$\dfrac{q-r}{p-q}$$$$p^{th}, \ q^{th}, \ r^{th}$$ terms of A.P. are $$a + (p-1)d = x$$ (1)$$a + (q-1)d = xR$$ (2)$$a + (r-1)d = xR^2$$ (3)Where r is common ratio of G.P. Subtracting (2) from (3) and (1) from (2) and then dividing the former by the later, we have$$\dfrac{q-r}{p-q} = \dfrac{xR^2 - xR}{xR - x} = R$$

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