Question

If p, q, r are in H.P and the (p + 1)th, (q + 1)th and (r + 1)th terms of an A.P. are in G.P., then the ratio of the first term to the common difference of the A.P. is equal to

(a) \(\frac{-q}{2}\)

(b) \(\frac{-pr}{q}\)

(c) \(\frac{-pr}{q^2}\)

(d) \(\frac{-2q}{pr}\)

Answer 1

(a) \(-\frac{q}{2}\)

p, q, r are in H.P. ⇒ q = \(\frac{2pr}{p+r}\)            .....(i)

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

∴ T_{p + 1} = a + pd 

T_{q + 1} = a + qd 

T_{r + 1} = a + rd 

Given, T_{p + 1}, T_{q + 1}, T_{r + 1} are in G.P. 

⇒ (T_{q + 1})² = T_{p + 1} · T_{r + 1} 

(a + qd)² = (a + pd) (a + rd) 

⇒ a² + 2aqd + q²d² = a² + ad (p + r) + prd² 

⇒ ad (2q – p – r) = d² (pr – q²)

⇒ \(\frac{a}{d}\) = \(\frac{pr-q^2}{2q-p-r}\) = \(\frac{\frac{1}{2}(p+r)q-q^2}{2q-(p+r)}\)          (Using (i))

= \(\frac{-\frac{q}{2}[2q-(p+r)]}{2q-(p+r)}\) = \(-\frac{q}{2}.\)