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Question

If a, b, c and d are in G.P. show that (a2+b2+c2)(b2+c2+d2)=(ab+bc+cd)2

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Solution

Taking LHS
(a2+b2+c2)(b2+c2+d2)
putting values b=ar,c=ar2,d=ar3
(a2+(ar)2+(ar2)2)((ar)2+(ar2)2+(ar3)2)
=a2(1+r2+r4)a2(r2+r4+r6)
=a4r2(1+r2+r4)2
RHS=(ab+bc+cd)2
=(a×ar+ar×ar2+ar2×ar3)2
=(a2r+a2r3+a2r5)2
=ar2(1+r2+r4)2
=LHS.

1156021_1196026_ans_52a42c4d5d40422d85d2db264109645a.jpg

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