# Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

$\begin{array}{1 1}(A)\;\sqrt 2+ \sqrt 3 \\ (B)\;3+\sqrt 2 \\(C)\;2- \sqrt 3 \\(D)\;2+ \sqrt 3 \end{array}$

Let the numbers be $a, ar, ar^2$
If $ar$ is doubled then,
$\implies 2ar = \frac{ar^2 + a}{2}$
$\implies 4ar = ar^2 + a$
$\implies 4r = r^2 + 1$
$\implies r^2 - 4r + 1$
On solving we get
$r = 2 \pm \sqrt 3$
Since it is an increasing
$G.P$ $'r' = 2 + \sqrt3$
edited Nov 7, 2017