Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence with a definite number of terms.

In General we could write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, … }

Where,

a is the first term, and
d is the difference between the terms (called the “common difference”)

Common Difference in Arithmetic Series:

Suppose, a1, a2, a3, ……., an is an AP, then; the common difference “ d ” can be obtained as;

d = a₂ – a₁ = a₃ – a₂ = … = a_n – a_{n – 1}

nth Term of an Arithmetic Series:

a_n = a₁ + (n − 1) × d

Where,

A₁ = First term
d = Common difference
n = number of terms

Example:

Find an equation for the general term of the given arithmetic sequence and use it to calculate its 9^th term: 3, 8, 13, 18, 23, 28, 33, 38, …

Solution:

The values of a and d are:

a₁ = 3 (the first term)
d = 8-3= 5 (the “common difference”)

Using the Arithmetic Sequence rule:

x_n = a₁ + d(n−1)

x_n = 3 + 5(n−1)

x_n = 3 + 5n − 5

x_n = 5n − 2

So the 9th term is:

x₉ = 5×9 − 2

x₉ = 43

Sum of N Terms of Arithmetic Series

S_n = n/2[2a₁ + (n − 1) × d]

Example:

Find the sum of the first 15 terms of the arithmetic sequence 1,2,3,4,…

Solution:

S = n/2[2a₁ + (n − 1) × d]

S = = 15/2[2.1+(15-1).1]

S = 15/2[2+14]

S = 15/2 [16] = 15 x 8

S = 120

Sum of Arithmetic Series when the Last Term is Given

S_n = n/2 (a₁+a_n)