2015/017) If the (m+1)th,(n+1)th,& (r+1)the terms of an A.P are in G.P & m,n,r are in H.P .Show that the ratio of the common difference to the first term in the A.P is -2/n

Let 1^st term be a and difference be d

so (m+1)st term a + md

(n+1)st term a + nd

(r+1) st term a + rd

these are in GP

so $(a+md)(a+rd) = (a+nd)^2$

or $(m+r) ad + rmd^2 = 2nad + n^2d^2$

or $ad(m+r – 2n) = d^2(n^2 – rm)$

or $\dfrac{d}{a} = \dfrac{m+ r – 2n}{n^2-rm}$

as m,n,r are in HP so $\dfrac{1}{m} + \dfrac{1}{r} = \dfrac{2}{n}$

or $r + m = \dfrac{2rm}{n}$

so from (1)

$\dfrac{d}{a} = \dfrac{\frac{2rm}{n} – 2n}{n^2-rm} =\dfrac{-2}{n}$