## Arithmetic Series

### 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 = … = an – an – 1

### nth Term of an Arithmetic Series:

an = 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 9th 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:

xn = a1 + d(n−1)

xn = 3 + 5(n−1)

xn = 3 + 5n − 5

xn = 5n − 2

So the 9th term is:

x9 = 5×9 − 2

x9 = 43

### Sum of N Terms of Arithmetic Series

Sn = 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  = 15 x 8

S = 120

Sn = n/2 (a1+an)