# Geometric Progression

A sequence of non-zero numbers is called geometric progression. The abbreviated form is G.P. In G.P. the ratio of a term and the term preceding to it is always same or constant. The constant ratio is called common ratio of the G.P.## Definition of Geometric Progression

Let a sequence $a_{1},a_{2},a_{3},a_{4},a_{5},,...,a_{n}$,... is called a G.P if$\frac{a_{n+1}}{a_{n}}$ = constant for all n $\epsilon $ N.

**Example 1 :**The sequence 6,18,54,162,... is a G.P because

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

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

**Example 2 :**The sequence $\frac{1}{3},\frac{-1}{2},\frac{3}{4},\frac{-9}{8}$,,... is a G.P. because

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

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

**Example 3 :**4,-2,1,$\frac{-1}{2}$,...is a G.P because

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

**Example 4 :**Show that the sequence given by $a_{n} = 3(2^{n})$, for all

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

**Solution :**We have $a_{n} = 3(2^{n})$

∴ $a_{n+1} = 3(2^{n+1})$

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.

**Geometric series**If $a_{1},a_{2},a_{3},a_{4},...a_{n}$is a G.P the the expression $a_{1}+ a_{2} + a_{3}+....+a_{n}... is called geometric series.

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

**Practice questions**

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,...

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