Derivation of Sum of Finite and Infinite Geometric Progression

Geometric Progression, GP
Geometric progression (also known as geometric sequence) is a sequence of numbers where the ratio of any two adjacent terms is constant. The constant ratio is called the common ratio, r of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by r.
 

Eaxamples of GP:

• 3, 6, 12, 24, … is a geometric progression with r = 2
• 10, -5, 2.5, -1.25, … is a geometric progression with r = -1/2

The n^th term of geometric progression
Given each term of GP as a₁, a₂, a₃, a₄, …, a_m, …, a_n, expressing all these terms according to the first term a₁ will give us...
$a_1 = a_1$

$a_2 = a_1 r$

$a_3 = a_2 r = (a_1 r) \, r = a_1 r^2$

$a_4 = a_3 r = (a_1 r^2) \, r = a_1 r^3$

$\dots$

$a_m = a_1 r^{\, m - 1}$

$\dots$

Where
a₁ = the first term, a₂ = the second term, and so on
a_n = the last term (or the n^th term) and
a_m = any term before the last term
 

Sum of Finite Geometric Progression
The sum in geometric progression (also called geometric series) is given by
$S = a_1 + a_2 + a_3 + a_4 + \, \dots \, + a_n$

$S = a_1 + a_1 r + a_1 r^2 + a_1 r^3 + \, \dots \, + a_1 r^{\, n - 1}$   ←   Equation (1)
 

Multiply both sides of Equation (1) by r will have
$Sr = a_1 r + a_1 r^2 + a_1 r^3 + a_1 r^4 + \, \dots \, + a_1 r^{\, n}$   ←   Equation (2)
 

Subtract Equation (2) from Equation (1)
$S - Sr = a_1 - a_1 r^{\, n}$

$(1 - r)S = a_1 (1 - r^{\, n})$

The above formula is appropriate for GP with r < 1.0
 

Subtracting Equation (1) from Equation (2) will give
$Sr - S = a_1 r^{\, n} - a_1$

$(r - 1)S = a_1 (r^{\, n} - 1)$

This formula is appropriate for GP with r > 1.0.
 

Sum of Infinite Geometric Progression, IGP
The number of terms in infinite geometric progression will approach to infinity (n = ∞). Sum of infinite geometric progression can only be defined at the range of -1.0 < (r ≠ 0) < +1.0 exclusive.
 

From
$S = \dfrac{a_1 (1 - r^{\, n})}{1 - r}$

$S = \dfrac{a_1 - a_1 r^{\, n}}{1 - r}$

$S = \dfrac{a_1}{1 - r} - \dfrac{a_1 r^{\, n}}{1 - r}$
 

For n → ∞, the quantity (a₁rⁿ) / (1 - r) → 0 for -1.0 < (r ≠ 0) < +1.0, thus,

See also the derivation for the product of GP.