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If a,b,c,d are in G.P., prove that (a^2+...

If a,b,c,d are in G.P., prove that `(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2.`

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Let a,b,c,d are in G.P. Let the common ratio=r `impliesb=ar,c=ar^2,d=ar^3` LHS=`(a^2+b^2+c^2)(b^2+c^2+d^2)=(a^2+a^2r^2+a^2r^4)(a^2r^2+a^2r^4+a^2r^6)=a^4r^2(1+r^2+r^4)^2=(a^2r+a^2r^3+a^2r^5)^2(a.ar+ar.ar^2+ar^2.ar^3)^2=(ab+bc+cd)^2=R.H.S.`(proved)
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