If (m+1)^ th , (n+1)^ th , and (r+1)^ th terms of a non-constant A.P. are in G.P. and m, n, r are in H.P., then the ratio of first term of the A.P to its common difference is:

Step-by-step solution

Let the A.P. be A, A+D, A+2D, The terms are T_ m+1 = A + mD , T_ n+1 = A + nD , T_ r+1 = A + rD . Given they are in G.P., (A + nD)^2 = (A + mD)(A + rD) . A^2 + 2AnD + n^2D^2 = A^2 + A(m+r)D + mrD^22AnD - A(m+r)D = mrD^2 - n^2D^2A(2n - m - r)D = (mr - n^2)D^2 Since the A.P. is non-constant, D 0 . A D = mr - n^2 2n - m - r Since m, n, r are in H.P., n = 2mr m+r m+r = 2mr n . Substitute m+r into the equation: A D = mr - n^2 2n - 2mr n = n(mr - n^2) 2n^2 - 2mr = n(mr - n^2) -2(mr - n^2) = - n 2