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

Answers (1)
S seema garhwal

 If a, b, c and d are in G.P.

bc=ad....................(1)

b^2=ac....................(2)

c^2=bd....................(3)

To prove : (a^2 + b^2 + c^2) (b^2 + c^2 + d^2) = (ab + bc + cd)^2 .

RHS : (ab + bc + cd)^2 .

       =(ab + ad + cd)^2 .

      =(ab + d (a+ c))^2 .

     =a^2b^2 + d^2 (a+ c)^2 + 2(ab)(d(a+c))

    =a^2b^2 + d^2 (a^2+ c^2+2ac) + 2a^2bd+2bcd

  Using equation (1) and (2),

  =a^2b^2 + 2a^2c^2+ 2b^2c^2+d^2a^2+2d^2b^2+d^2c^2

=a^2b^2 + a^2c^2+ a^2c^2+b^2c^2+b^2c^2+d^2a^2+d^2b^2+d^2b^2+d^2c^2

=a^2b^2 + a^2c^2+ a^2d^2+b^2.b^2+b^2c^2+b^2d^2+c^2b^2+c^2.c^2+d^2c^2

=a^2(b^2 + c^2+ d^2)+b^2(b^2+c^2+d^2)+c^2(b^2+c^2+d^2)

=(b^2 + c^2+ d^2)(a^2+b^2+c^2) = LHS

Hence proved

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