NIMCET 2017 — Mathematics PYQ

Concept:
If a, b, c are in GP then $\frac{\mathrm{b}}{\mathrm{a}}=\frac{\mathrm{c}}{\mathrm{b}}$
Calculations:

Consider, three positive number a-d,a, a+ d are in AP
Three positive number whose sum is 21.
$\Rightarrow$a-d+a+a+d=21

$\Rightarrow$a=7

Hence, three positive number becomes 7-d, 7, 7+ d.

It is given, If 2,2,14 are added to them respectively then resulting number are in geometric progression.

$\Rightarrow 9-$d,9,21+d are in GP.

$\Rightarrow \frac{9}{9-\mathrm{d}}=\frac{21+\mathrm{d}}{9}$

$$\Rightarrow 81=(21+d)(9-d)$$
$$<br>\Rightarrow d^2+12d-108=0$$
$$\Rightarrow (d+18)(d-6)=0$$
$$<br>\Rightarrow d=-18\text{ and }d=6$$
When $d=-18$, the three numbers are 25, 7, -11.
When $d=6$, the three numbers are 1, 7, 13.