A hidden GP in an AP!

Let the Arithmetic Progression be :

$a,a+d,a+2d,\ldots,a+7d$
where, $\displaystyle a+(n-1)d$ represents the $\displaystyle n^{th}$ term

According to the problem,

$\displaystyle a+d,a+5d,a+7d$ are in Geometric Progression.

Hence,

$(a+5d)^2 = (a+d)(a+7d)$

$\Rightarrow 9d+a = 0$

$\Rightarrow \frac{a}{d} = -9$

Keeping this in mind,
Common ratio $\displaystyle = r = \frac{a+5d}{a+d} = \frac{\frac{a}{d}+5}{\frac{a}{d}+1} = \frac{1}{2}$

Hence,
$\frac{a}{b} = \frac{1}{2}$

$\Rightarrow a+b= 3$

The Geometric Progression is $x+y, x+5y,$ and $x+7y$ , with both $x$ and $y$ as a real numbers. Easily, we could calculate the common ratio as $\frac{x+5y}{x+y} = \frac{x+7y}{x+5y} = \frac{2y}{4y} = \frac{1}{2}$ . Therefore, $a+b = 3$ .