Arithmetic and Geometric Series Explained with Formulas and Examples

In math, we often see patterns made from numbers. Two common types of patterns are Arithmetic and Geometric Series. First, an arithmetic series is a pattern where the same number is added each time. On the other hand, a geometric series is formed by multiplying by the same number each time. Because of these patterns, we can solve many real-life problems, such as calculating savings, interest, or population growth. Overall, understanding these series makes math more helpful and easier to use.

Arithmetic Series

An Arithmetic Series is the sum of the terms in an Arithmetic Sequence, where each term increases by a constant value called the common difference d.

General Form of an Arithmetic Sequence

a, a + d, a + 2d, a + 3d, ……….. a+(n-1)d

Example

2, 5, 8, 11, 14,…….

Here:

• First term a = 2
• Common difference d = 3

Each term increases by 3:

5 – 2 = 3, 8 – 5 = 3, and so on

Sum of an Arithmetic Sequence

To find the sum of the first n terms:
S_n = (n/2) *( (2*a + (n – 1)*d))

Example: find the sum of first 4th term of series 2, 5, 8, 11, 14,…….

Given a = 2, d = 3, and n = 4:

S_n =(4/2)*((2*2+(4-1)*3))

S_n = 2*(4+9)

S_n = 26

Geometric Series

A Geometric Series is the sum of the terms in a Geometric Sequence, where each term is multiplied by a constant called the common ratio r.

General Form of Geometric Series

A Geometric Series is formed by repeatedly multiplying by a constant ratio r:

a+ar+ar²+ar³+⋯+arⁿ⁻¹

Example

3, 6, 12, 24, 48,…………….

Here:

• First term a = 3
• Common ratio r = 2

• a is the first term,
• r is the common ratio,
• n is the number of terms (if finite).

Sum of Geometric Sequence

To find the sum of the first n terms (for r is not equal to 1):
S_n = a*(1 – rⁿ)/(1 – r)

Example: Find Sum of First 4 Terms of series 3, 6, 12, 24, 48,…………….

Given a = 3, r = 2, and n = 4:

S_n = 3* (1-2^4)/(1-2)

S_n = 3*(-15)/-1

S_n =-45/-1

S_n =45

Summary