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Grade 11Algebra

if (m+1)th, (n+1)th and (r+1)th terms of ap are in gp and m, n, r are in HP, then show that the ratio of the common difference to the first term in the ap is ( -2/n).

Profile image of Samuel Garry
8 Years agoGrade 11
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1 Answer

Profile image of Arun
ApprovedApproved Tutor Answer8 Years ago
Let 1st term be a and common difference be t
(m+1)st term = a + mt
(n+1) st term = a + nt
(r+1s)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
 
t = -2/n

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