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# The sum of three numbers in GP is 26. If 2,2,-6 are added to them respectively they form as.n AP . Find the number

$(a)\;2,6,12\qquad(b)\;18,6,2\qquad(c)\;2,4,6\qquad(d)\;2,8,16$

Can you answer this question?

Answer : (b) 18,6,2
$Explanation \;:\; Let\;the\;three\;numbers\;in\;GP\;be\;a\;,ar\;,ar^2$
$a+ar+ar^2=26\qquad\;-----------(1)$
$Given\; that\;a+2\;,ar+2\;,ar^2-6\;\;are \;in\;AP$
$=2(ar+2)a+2+ar^2-6$
$-a+2ar-ar^2=-8\;-----------(2)$
$Adding \;equations\;(1)\;\xi\;(2)$
$a+ar+ar^2\;=\;26$
$-a+2ar-ar^2=\;-8\;$
$3ar\;=\;18$
$ar=6\quad\;,a=6/r$
$substituting\;in\;equation\;(1)$
$6/r+6+6r=26$
$3r^2-10r+3=0$
$r=1/3\quad\;or\quad\;a=18\quad\;or\;\quad2$
$GP\;is\;2,6,18\quad\;or\quad\;18,6,2.$
answered Jan 3, 2014 by