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, \dots \, is a geometric progression with r = 2
• 10, \, -5, \, 2.5, \, -1.25, \dots \, is a geometric progression with r = -\frac{1}{2}

The n^th term of geometric progression

Given each term of GP as a_1, \, a_2, \, a_3, \, a_4, \, \dots \, a_m, \, \dots \, a_n \,, expressing all these terms according to the first term a_1 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
a_1 = the first term, a_2 = 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} \, \to \, 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} \, \to \, 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 = \infty). Sum of infinite geometric progression can only be defined at the range of \, -1.0 < r < +1.0 \, inclusive.

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