What is the common ratio in Geometric Progression?

A branch of mathematics that deals usually with the nonnegative real numbers including sometimes the transfinite cardinals and with the appliance of the operations of addition, subtraction, multiplication, and division to them. The basic operations under arithmetic are addition, subtraction, division, and multiplication. The operations are done using the BODMAS rule. The BODMAS rule is followed to order any operation involving +, −, ×, and ÷. The order of operation is, B – Brackets, O – Order, D – Division, M – Multiplication, A – Addition, and S – Subtraction.

Progression

Progression may be a list of numbers (or items) that exhibit a specific pattern. The difference between a sequence and a progression is that to calculate its nth term, a progression has a specific formula i.e. T_n = a + (n-1)d  which is the formula of nth term of an arithmetic progression. There are generalized formulae for other types of progressions too, like geometric progression, harmonic progression, etc.

Geometric Progression

If in a sequence of terms, each succeeding term is generated or obtained by multiplying each preceding term with a constant or fixed value, then the sequence is called a geometric progression. (GP), whereas the constant value or fixed value is called the common ratio and usually it is represented by ‘r’.

What is a Common Ratio?

In geometric progression, the common ratio is the ratio between any term in the sequence and divided by the previous term.

The Formula to calculate the common ratio in geometric progression, a, ar, ar², ar³, ar⁴, ar⁵… is,

Common ratio = ar/ a = ar²/ ar = ……. = a_n/ a_n-1

As the definition states , we can calculate the common difference of an geometric progression by dividing any term by its previous term.

Sample Problems

Question 1: 3, 9, 27, 81, 243, 729, … is a GP, where the common ratio is?

Solution:

It can be calculated as 

27/ 9 = 3 or 9/ 3 = 3 

Question 2: Consider the following series:  1/2, 1/4, 1/8, 1/16, 1/32, … Find the Common ratio?

Solution: 

The common ratio is 

(1/4) ÷ (1/2) = (1/4 ) × (2/1 ) = 2/4 = 1/2

Question 3: Consider the following series :  0.2 , 0.6 , 1.8 , 5.4 , 16.2 ,… Find the Common ratio.

Solution:

The common ratio is 

1.8 /0.6 = 3 

Question 4: Can the Common ratio of any geometric progression be negative or not?

Answer:

Yes, common ratio of any geometric progression can be negative. For example, G.P= -16, 8, -4, 2, -1…

The common ratio is -16/8= -2.

Question 5: If first term and common ratio are given, can the whole geometric progression series be constructed?

Answer:

Yes, we can construct the whole series if the common ratio and first term are given.

For instance: First term= 40, Common ratio= 2

Next term= 40/2= 20, next term= 20/2= 10

Similarly, GP= 40, 20, 10, 5,…