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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
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