if `(m+1)th, (n+1)th and (r+1)th` term of an AP are in GP.and m, n and r in HP. . find the ratio of first term of A.P to its common difference

Explanation

Let a be the first term and d be the common difference of the given A.P. Then, as given the (m+1)th,(n+1)th, and (r+1)th terms are in G.P. So,
a+md,a+nd,a+rd are in G.P.
`rArr(a+nd)^(2)=(a+md)(a+rd)`
`rArra(2n-m-r)=d(mr-n^(2))`
`rArrd/a=(2n-(m+r))/(mr-n^(2))` (1)
Next, m ,n r are in H.P. Hence,
`n=(2mr)/(m+r)` (2)
From (1) and (2),
`d/a(2n-(m+r))/(mr-n^(2))=2/n((2n-(m+r))/((m+r)-2n))=-2/n`