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# If a , b , c are in AP and A.G are arithmetic and geometric mean , between a and b while $\;A^{|}\;and\;G^{|}$ are A.M and G.M between B and C . then

$(a)\;A^2+G^2={A^{|}}^{2}+{G^{|}}^{2}\qquad(b)\;A^2-G^2={A^{|}}^{2}-{G^{|}}^{2}\qquad(c)\;A^2+{A^{|}}^{2}=G^2+{G^{|}}^{2}\qquad(d)\;{A^{|}}^{2}+A^2={G^{|}}^{2}-G^2$

Answer : (b) $A^2-G^2={A^{|}}^{2}-{G^{|}}^{2}$
Explanation : $\;A^{2}=(\large\frac{a+b}{2})^2\quad\;G^2=ab$
$A^2-G^2=\frac{1}{4}\;(a-b)^2=\frac{1}{4}\;d^2$
Similarly ,
${A^{|}}^{2}-{G^{|}}^{2}=\frac{1}{4}\;(b-c)^2=\frac{1}{4}\;d^2\;.$