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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

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Solution

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$$p^{th}, \ q^{th}, \ r^{th}$$ 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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