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 β−α6αβ(β4−α4) equals to
A
23
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B
32
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C
43
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D
34
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Solution
The correct option is B32 α,β,a,b are in A.P b=a+3(β−α)=3β−2α α,β,c,d are in G.P. d=α(βα)3=β3α2 α,β,e,f are in H.P 1f=1α+3(1β−1α) f=αβ3α−2β Now since b, d, f are in G.P. d2=bf (β3α2)2=(3β−2α)(αβ)3α−2β β5(3α−2β)=α5(3β−2α) 3αβ(β4−α4)=2(β6−α6) 32=β6−α6αβ(β4−α4)