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Question

Given that α,β,a,b are in A.P., α,β,c,d are in G.P. and α,β,e,f are in H.P. If b,d,f are in G.P., then the value of 2(α6β6)αβ(α4β4),0<α<β is

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Solution

Given that α, β, a, b are in A.P.
bα=3(βα)b=3β2α....(1)

Also, α, β, c, d are in G.P.
dα=(βα)3d=β3α2.....(2)

Similarly, using the last condition, we get
1f=3β2α......(3)

Given that b,d,f are G.P.,
d2=bfβ6α4=αβ(3β2α3α2β)β5(3α2β)=α5(3β2α)2(α6β6)=3αβ(α4β4)
So, 2(α6β6)αβ(α4β4)=3

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