Show that the progressions are GP. Also, find the common ratio.
$4, - 2, 1, - 1 / 2, \dots$

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Solution

Examine the progression is GP and find the common ratio:
The sequence is $4, - 2, 1, - 1 / 2, \dots$
Here, $a_{1} = 4, a_{2} = - 2, a_{3} = 1$
$\Rightarrow \frac{a_{2}}{a_{1}} = \frac{- 2}{4} = - \frac{1}{2}$
$\Rightarrow \frac{a_{3}}{a_{2}} = \frac{1}{- 2}$
$\Rightarrow \frac{a_{4}}{a_{3}} = \frac{1}{- 2} \times 1 = - \frac{1}{2}$ and so on.
Since the ratio of any term by its preceding term is constant.
Hence, the sequence $4, - 2, 1, - 1 / 2, \dots$ is GP and the common ratio $r = - \frac{1}{2}$ .

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