# 1.4.1 Characteristics of a Geometry Progressions

2. Geometric Progression
(A) Characteristics of Geometric Progressions
Geometric progression is a progression in which the ratio of any term to the immediate term before is a constant. The constant is called common ratio, r.

Example:
Determine whether or not each of the following number sequences is a geometric progression (GP).
(a) 1, 4, 16, 64, …..
(b) 10, –5, 2.5, –1.25, …..
(c) 2, 4, 12, 48, …..
(d) –6, 1, 8, 15, …..

Solution:
[Smart Tips: For GP, times a fixed number every time to get the next time.]

(a)

$\begin{array}{l}\text{Common ratio,}r=\frac{{T}_{n}}{{T}_{n-1}}\\ \frac{{T}_{3}}{{T}_{2}}=\frac{16}{4}=4,\text{}\frac{{T}_{2}}{{T}_{1}}=\frac{4}{1}=4\\ \frac{{T}_{3}}{{T}_{2}}=\frac{{T}_{2}}{{T}_{1}}\end{array}$
This is a GP, a =1, r = 4.

(b)

$\begin{array}{l}\text{Common ratio,}\\ r=\frac{{T}_{2}}{{T}_{1}}=-\frac{5}{10}=-\frac{1}{2}\end{array}$
This is a GP, a =1, r = – ½.

(c)

This is NOT a GP.
This is because the ratio of each term to its preceding term is not a similar constant.

(B) The steps to prove whether a given number sequence is a geometric progression.
Step 1: List down any three consecutive terms. [Example: T1, T2, T3.]
Step 2: Calculate the values of  $\frac{{T}_{3}}{{T}_{2}}\text{and}\frac{{T}_{2}}{{T}_{1}}.$
Step 3: If  $\frac{{T}_{3}}{{T}_{2}}=\frac{{T}_{2}}{{T}_{1}}=r$ , then the number sequence is a geometric
progression.
Step 4: If  $\frac{{T}_{3}}{{T}_{2}}\ne \frac{{T}_{2}}{{T}_{1}}$ , then the number sequence is not a geometric progression.