Geometric Progression

A geometric progression is a sequence in which each term (after the first) is determined by multiplying the preceding term by a constant. This constant is called the common ratio of the arithmetic progression. A geometric progression can be defined as follows:

The geometric progression { a_n } = a₁, a₂, a₃, ...., a_n , ....
where n = 1, 2, 3, ....

Its terms are determined by the equation a_n = a , where

a₁ is the first term of the geometric progression
a_n is the nth term of the geometric progression
n is the term number
r is the common ratio of the geometric progression

The sum of the first n terms of an geometric progression is calculated as

S_n = ( a₁ - a₁ rⁿ ) / ( 1 - r)

or

S_n = ( a₁ - a_n r ) / ( 1 - r) where a_n r = a1 · r^n-1 = a₁ rⁿ

EX. In the sequence { a_n } = 1, 2, 4, 8, 16 , ...., where n = 1, 2, 3, ....

a_n = (1) 2^n-1= 2^n-1

Thus, the sequence { a_n } = 1, 2, 4, 8, 16 , .... is a geometric progression with a₁ = 1 and r = 2. The 6th to 10th terms of this geometric sequence is

a₆ = 2^6-1 = 2⁵ = 32
a₇ = 2^7-1 = 2⁶ = 64
a₈ = 2^8-1 = 2⁷ = 128
a₉ = 2^9-1 = 2⁸ = 256
a₁₀ = 2^10-1 = 2⁹ = 512

The sum of the first n terms of the sequence { a_n } = 1, 2, 4, 8, 16 , . . is

S_n = (1 - (1) 2_n ) / (1 - 2) = (1 - 2ⁿ) / (-1) = 2ⁿ - 1

We can verify this for the first 5 terms:

S₁ = 2¹- 1 = 1
S₂ = 2²- 1 = 1 + 2 = 3
S₃ = 2³- 1 = 1 + 2 + 4 = 7
S₄ = 2⁴- 1 = 1 + 2 + 4 + 8 = 15
S₅ = 2⁵- 1 = 1 + 2 + 4 + 8 + 16 = 31