Find $x$ , so that $x, x + 2, x + 6$ are consecutive terms of geometric progression.

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Solution

The ratio between consecutive terms is constant in geometric progression
$\Rightarrow \frac{x + 2}{x} = \frac{x + 6}{x + 2}$
$\Rightarrow (x + 2)^{2} = x (x + 6)$
$\Rightarrow x^{2} + 4 x + 4 = x^{2} + 6 x$
$\Rightarrow 2 x = 4$
$\Rightarrow x = 2$

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