Show that one of the following progression is a G.P. Also, find the common ratio in case:

4, −2, 1, −1/2, ...

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#### Solution

We have,

\[ a_1 = 4, a_2 = - 2, a_3 = 1, a_4 = - \frac{1}{2}\]

\[\text { Now }, \frac{a_2}{a_1} = \frac{- 2}{4} = \frac{- 1}{2}, \frac{a_3}{a_2} = \frac{1}{- 2}, \frac{a_4}{a_3} = \frac{- \frac{1}{2}}{1} = \frac{- 1}{2}\]

\[ \therefore \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \frac{- 1}{2}\]

\[\text { Thus, } a_1 , a_2 , a_3 \text { and } a_4\text { are in G . P . , where a = 4 and }r = \frac{- 1}{2} .\]

Concept: Geometric Progression (G. P.)

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