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

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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

Posted by

seema garhwal

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