**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 a

_{1}, a

_{2}, a

_{3}, a

_{4}, …, a

_{m}, …, a

_{n}, expressing all these terms according to the first term a

_{1}will give us...

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

→ Equation (1)

Multiply both sides of Equation (1) by r will have

→ Equation (2)

Subtract Equation (2) from Equation (1)

The above formula is appropriate for GP with r < 1.0

Subtracting Equation (1) from Equation (2) will give

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

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

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