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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 59 of 106
Marks: +1, -0
If a, b, c and d are in G.P., show that
(a2+b2+c2)(b2+c2+d2)(a^2 + b^2 + c^2) (b^2 + c^2 + d^2) = (ab+bc+cd)2(ab + bc + cd)^2
Solution:  
We have a, b, c, d are in G.P.
Let r be a common ratio, then b = ar, c = ar2ar^2, d = ar3ar^3
∴ (a2+b2+c2)(b2+c2+d2)(a^2 + b^2 + c^2) (b^2 + c^2 + d^2)
= [a2+(ar)2+(ar2)2][a^2 + (ar)^2 + (ar^2)^2] [(ar)2+(ar2)2+(ar3)2][(ar)^2 + (ar^2)^2 + (ar^3)^2] = [a2+a2r2+a2r4][a^2 + a^2r^2 + a^2r^4] [a2r2+a2r4+a2r6][a^2r^2 + a^2r^4 + a^2r^6]
= a2(1+r2+r4)a2r2(1+r2+r4)a^2(1 + r^2 + r^4) a^2 r^2 (1 + r^2 + r^4)
= a4r2(1+r2+r4)2a^4 r^2 (1 + r^2 + r^4)^2 ....(i)
Also, (ab+bc+cd)2(ab + bc + cd)^2 = (a2r+a2r3+a2r5)2(a^2r + a^2r^3 + a^2r^5)^2
= a4r2(1+r2+r4)2a^4r^2 (1 + r^2 + r^4)^2 ....(ii)
From (i) & (ii), we get (a2+b2+c2)(b2+c2+d2)(a^2 + b^2 + c^2) (b^2 + c^2 + d^2) = (ab+bc+cd)2(ab + bc + cd)^2.
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