1.5 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)
Common ratio, r= T n T n1 T 3 T 2 = 16 4 =4,  T 2 T 1 = 4 1 =4 T 3 T 2 = T 2 T 1
This is a GP, a =1, r = 4.

(b)
Common ratio,  r= T 2 T 1 = 5 10 = 1 2
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  T 3 T 2  and  T 2 T 1 .    
Step 3: If  T 3 T 2 = T 2 T 1 =r , then the number sequence is a geometric 
             progression.
Step 4: If  T 3 T 2 T 2 T 1 , then the number sequence is not a geometric progression.