Geometric Progressions

A geometric progression is a sequence of numbers in which each successive number is obtained by multiplying a fixed quantity with the previous number.

The fixed or constant quantity that is multiplied to get the next term in a geometric progression is known as the common ratio. It is commonly represented by the letter r.

Let each term from the beginning in a geometric progression be represented by the following:

a₁, a₂, a₃, a₄, a₅, ... 

(where a₁ is the first term, a₂ the second term, and so on.)

In the above sequence, the second term (a₂) is obtained by multiplying a fixed quantity, the common ratio, to the first term (a₁). That is, if the common ratio is r,

a₂ = a₁ x r

Similarly the third term (a₃) is obtained by multiplying a fixed quantity (the common ratio) to the second term (a₂). That is, if the common ratio is r,

a₃ = a₂ x r

Thus the common ratio (r) of a geometric progression can be obtained by dividing its any term by the previous term. Thus, in the following geometric progression:

2, 4, 8, 16, ...

the common ratio is 4/2 = 2. Similarly the common ratios of the following geometric progressions are given:

• 3, 9, 27, ... (common ratio, r = 3)
• 2, 6, 18, ... (common ratio, r = 2)
• 1/2, 1/4, 1/8, 1/16, ... (common ratio, r = 1/2) 

Thus, if in a geometric progression, a is the first term and r is the common ratio, then

• Second term = a x r = ar
• third  term = (second term) x r = ar²
• fourth term = (third term) x r = ar³
• and so on.. 

Thus a geometric progression with first term a and common ratio r can be represented by

ar, ar², ar³, ar⁴, ... 

Related posts:

• Sum of 'n' terms of a geometric progression
• Sum of an infinite geometric progression 
• Progressions in general