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, [math]r[/math] of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by [math]r[/math].

Eaxamples of GP:

• [math]3, \, 6, \, 12, \, 24, \dots \,[/math] is a geometric progression with [math]r = 2[/math]
• [math]10, \, -5, \, 2.5, \, -1.25, \dots \,[/math] is a geometric progression with [math]r = -\frac{1}{2}[/math]

The n^th term of geometric progression

Given each term of GP as [math]a_1, \, a_2, \, a_3, \, a_4, \, \dots \, a_m, \, \dots \, a_n \,[/math], expressing all these terms according to the first term [math]a_1[/math] will give us...
[math]a_1 = a_1[/math]
[math]a_2 = a_1 r[/math]
[math]a_3 = a_2 r = (a_1 r) \, r = a_1 r^2[/math]
[math]a_4 = a_3 r = (a_1 r^2) \, r = a_1 r^3[/math]
[math]\dots[/math]
[math]a_m = a_1 r^{\, m - 1}[/math]
[math]\dots[/math]

Where
[math]a_1[/math] = the first term, [math]a_2[/math] = the second term, and so on
[math]a_n[/math] = the last term (or the n^th term) and
[math]a_m[/math] = any term before the last term

Sum of Finite Geometric Progression

The sum in geometric progression (also called geometric series) is given by
[math]S = a_1 + a_2 + a_3 + a_4 + \, \dots \, + a_n[/math]
[math]S = a_1 + a_1 r + a_1 r^2 + a_1 r^3 + \, \dots \, + a_1 r^{\, n - 1} \, \to \, [/math] Equation (1)

Multiply both sides of Equation (1) by [math]r[/math] will have
[math]Sr = a_1 r + a_1 r^2 + a_1 r^3 + a_1 r^4 + \, \dots \, + a_1 r^{\, n} \, \to \, [/math] Equation (2)

Subtract Equation (2) from Equation (1)
[math]S - Sr = a_1 - a_1 r^{\, n}[/math]
[math](1 - r)S = a_1 (1 - r^{\, n})[/math]

The above formula is appropriate for GP with [math]r < 1.0[/math]

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

This formula is appropriate for GP with [math]r > 1.0[/math]

Sum of Infinite Geometric Progression, IGP

The number of terms in infinite geometric progression will approach to infinity ([math]n = \infty[/math]). Sum of infinite geometric progression can only be defined at the range of [math]\, -1.0 < r < +1.0 \,[/math] inclusive.

From
[math]S = \dfrac{a_1 (1 - r^{\, n})}{1 - r}[/math]
[math]S = \dfrac{a_1 - a_1 r^{\, n}}{1 - r}[/math]
[math]S = \dfrac{a_1}{1 - r} - \dfrac{a_1 r^{\, n}}{1 - r}[/math]

For [math]n \to \infty[/math], the quantity [math]\dfrac{a_1 r^{\, n}}{1 - r} \to 0[/math] for [math]\, -1.0 < r < +1.0 \,[/math], thus,