Q13/053) If the (m+1)th,(n+1)th,& (r+1)the terms of an A.P r 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 common difference be t

(m+1)st term = a + mt

(n+1) st term = a + nt

(r+1^s)st term = a + rt

As they are in GP

So (a+ mt)(a+rt) = (a+nt)^2

=> a^2 + a (m+r) t + mrt^2 = a^2 + 2ant + n^2 t^2

=> a (m+r) t + mrt^2 = 2ant + n^2 t^2

 => a (m+r)  + mrt  = 2an  + n^2 t

=> a(m+r -2n) = t(n^2- mr)

=>t/a = (n^2 – mr)/(m+r -2n) ..(1)

As m .n and r are in HP 1/m + 1/r = 2/n ot (m+r ) = 2mr/n

So m+r – 2n = (2mr/n- 2n) = 2/n(rm-n^2) …(2)

From (1) and (2) we get the result