Geometric Progressions

Geometric Progressions

1. Geometric progression: A sequence is called a geometric progression (G.P) if the ratio of any two consecutive terms is the same.
Examples:i)\( 1,\frac{1} {2},\frac{1} {4},\frac{1} {8},..... \)
                 ii)\( 3,3^2 ,3^3 ,3^4 ,..... \)
                 iii)\( \frac{{ - 1}} {3},\frac{1} {9},\frac{{ - 1}} {{27}},\frac{1} {{81}},... \)

2. The general form of the terms of a G.P. are \( a,ar,ar^2 ,...... \) where ‘a’ is the first term and ‘r’ is the common ratio.

3.  If ‘a’ is the first term and ‘r’ is the common ratio, then the general term or nth term or last term of a G.P. is
       \( t_n = ar^{n - 1} \) 
Explanation: Consider a G.P. 2, 6, 18, 54, ......
    here the first term = a = 2
    common ratio \( = r = \frac{{t_2 }} {{t_1 }} = \frac{6} {2} = 3 \)
Now, first term \( t_1 = 2 = a \)
        Second term \( t_2 = 6 = 2 \times 3 = ar \) 
        Third term \( t_3 = 18 = 2.3^2 = a.r^2 \)
        Fourth term \( t_4 = 54 = 2.3^3 = a.r^3 \)
Proceeding in this way, we get \( n^{th} \)term or last term \( = t_n = a.r^{n - 1} \)

4.     The common ratio of a G.P. \( r = \frac{{t_n }} {{t_{n - 1} }} \)

5.     The \( n^{th} \)term from the end of a finite G.P with last term l and common ratio r is \( \frac{l} {{r^{n - 1} }} \)

6.     Properties of G.P
i)  a, b, c are in G.P \( \Leftrightarrow b^2 = ac \)
ii) In a finite G.P the product of the terms equidistant from the beginning and end is always same and is equal to the product of the first and last terms.
i.e., \( a_1 .a_n = a_2 .a_{n - 2} = a_3 .a_{n - 3} = ...... \)
iii) If \( a_1 ,a_2 ,a_3 ,.....a_{n - 2} ,a_{n - 1} ,a_n \) are in G.P, then \( a_n ,a_{n - 1} ,a_{n - 2} ,.....a_3 ,a_2 ,a_1 \) are also in G.P.
iv) \( \lambda a_1 ,\lambda a_2 ,\lambda a_3 ,........\lambda a_n \) are in G.P where \( \lambda \in R - \left\{ 0 \right\} \)
v)  \( a_1^n ,a_2^n ,a_3^n ,......a_n^n \) are in G.P for \( n \in R \)
vi) \( \frac{1} {{a_1 }},\frac{1} {{a_2 }},\frac{1} {{a_3 }},.....\frac{1} {{a_n }} \) are in G.P
vii) If\( a_1 ,a_2 ,a_3 ,......a_n \)  is a G.P of non zero, non negative terms then \( \log a_1 ,\log a_2 ,....\log a_n \) are in A.P and vice versa.

7.     Some facts about G.P 
        If \( a_1 ,a_2 ,a_3 ,......a_n \) and \( b_1 ,b_2 ,b_3 ,.....b_n \) are two G.P’s with common ratios \( \text{r}_\text{1} \text{ and r}_\text{2} \) respectively, then
i) \( a_1 \pm b_1 ,a_2 \pm b_2 ,a_3 \pm b_3 ,.....a_n \pm b_n \) are not in G.P
ii)\( a_1 b_1 ,a_2 b_2 ,a_3 b_3 ,....a_n b_n \) are in G.P, with common ration \( r_1 r_2 \) 
iii)\( \frac{{a_1 }} {{b_1 }},\frac{{a_2 }} {{b_2 }},\frac{{a_3 }} {{b_3 }},.....\frac{{a_n }} {{b_n }} \)      are in G.P., with common ratio \( \frac{{r_1 }} {{r_2 }} \)

8.     Selection of terms in G.P


9.     Geometric Mean (G.M) : If three numbers a, b, c are in G.P, then b is called G.M of a  and c.
Note: The G.M of a and b is \( \sqrt {ab} \) 
Explanation: Let the G.M of a and b be ‘x’
We have a, x, b are in G.P 
\(\therefore\)    Common ratio \( = r = \frac{x} {a} = \frac{b} {x} \)
\( \begin{gathered} \Rightarrow x^2 = ab \hfill \\ \Rightarrow x = \sqrt {ab} \hfill \\ \end{gathered} \)
Note: If a and b are two numbers of oppposite sign, then G.M. between them does not exist.