# If $a , b , c$ are three numbers in GP . and $\;a+x\;,b+x\;,c+x\;$ are in HP then $x$ equals

$(a)\;a\qquad(b)\;b\qquad(c)\;c\qquad(d)\;\large\frac{a+b+c}{3}$

Explanation : $\large\frac{2}{b+x}=\large\frac{1}{a+x}+\large\frac{1}{c+x}$
Explanation : $\large\frac{2}{b+x}=\large\frac{a+c+2x}{(a+x)(c+x)}$
$2\;(a+x)(c+x)=(b+x)(a+c+2x)$
$2\;(ac+ax+cx+x^2)=bc+2bx+ab+ax+cx+2x^2$
$(ac-bc-ab+ac)=(2b-a-c)\;x$
$=b\;(2b-a-c)=(2b-a-c)\;x\quad\;[b^2=ac]$
$x=b\;.$