$\frac{a_{n+1}}{a_{n}}$ = constant for all n $\epsilon $ N.

$\frac{18}{6} = \frac{54}{18} = \frac{162}{54}$ = 3 which is constant.

Here the sequence of G.P with first term 6 and common ratio 3.

$\frac{-1}{2} \div \frac{1}{3} = \frac{3}{4} \div \frac{-1}{2} = \frac{-3}{2}$ which is constant

Here the sequence of G.P with first term $\frac{1}{3}$ and common ratio $\frac{-3}{2}$.

(-2) $\div (4) = (1) \div (-2) = \frac{-1}{2}$ which is constant.

n $\epsilon $ N, is a G.P. Also, find its common ratio.

∴ $a_{n+1} = 3(2^{n+1})$

So, $\frac{a_{n+1}}{a_{n}} = \frac{3(2^{n+1})}{3(2^{n})}$

$\frac{a_{n+1}}{a_{n}}$ = 2 which a constant for all n $\epsilon $N.

So the given sequence is a G.P. with common ratio 2.

Note that the geometric series is finite or infinite according to as the corresponding G.P. consists of finite or infinite number of terms.

I. Check whether the following sequence are in G.P. or not.

(i) -5, 15, -45, 135, ...

(ii) 0.5, 3.5, 24.5, 171.5, ...

(iii) 2, 6, 16, 54, ...

(iv) -11, 22, -44, 88, ...

(iv) 1/2, 1, 2, 4, 8,...

(v) −1, 1, 4, 8, ...

II. The following sequence is in G.P , find the first and the common ratio.

(i) −1, 6, −36, 216, ...

(ii) −3, −15, −75, −375, ...

(iv) 2, $\frac{1}{2}, \frac{1}{8},\frac{1}{128}, \frac{1}{512}$,...

(v) −24, −144, −864,...

From Geometric progression to Home

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