if (m+1)th,(n+1)th and (r+1)th terms of an A.P are in G.P,m,n&r are in H.P.then find the ratio of first term and common difference if this A.P.

Tm+1 = a + md

Tn+1 = a + nd

 Tr+1 = a + rd

Tm+1,Tn+1,Tr+1 are in gp therefore

(Tn+1)²  = (Tn+1)(Tr+1)

 (a+nd)² = (a+md)(a+rd)

 (a/d +n)^{2 =} (a/d + m)(a/d+ r)             ...............1

now   m,n,r are in hp therefore

 2/n =1/r + 1/m

 mr=n(m+r)/2.........2

let Z is the ratio of first term and common difference of AP then Z=a/d

putting Z in eq 1

 Z=mr - n² /2n -(m+r) 

after putting value of mr from eq 2 we get

Z=a/d = -n/2