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,...

11th grade math

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