Arithmetic and Geometric Series Explained with Formulas and Examples

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 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