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Question

If a, b, c are in G.P. then prove that : a2+ab+b2bc+ca+ab=b+ac+b

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Solution

Since, a, b, c are in G.P.

a = a, b = ar, c=ar2

a2+ab+b2bc+ca+ab=b+ac+b

a2+a(ar)+a2r2(ar)(ar)+(ar2)+a+(ar)=ar+aar2+ar

a2+(1+r+r2)a2(r3+r2+r)=a(1+r)a(r2+r)

1+r+r2r(1+r+r2)=1+r1+r)

1r=1r

LHS = RHS

Hence,

a2+ab+b2bc+ca+ab=b+ac+d


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