Arithmetic Sequences

Understanding Arithmetic sequences and series is a crucial part of studying mathematics as it is applicable in various fields such as computer programming, finance, statistics and physics. A sequence, also called a progression, is defined as the arrangement of individual terms in an orderly manner. These individual terms of a sequence, when added together, give rise to a series. Therefore, a series is the sum of terms in a progression. The order of elements or terms is important in a sequence but not in a series. These sequences and series can be classified into arithmetic progression.

Let us now discuss the series and sequence formula.

What are Arithmetic Sequences?

Let’s take a series: 3,7,11,15,19……..

In the given series, there is a pattern that can be seen which is that the difference between every two consecutive numbers is 4 {[7-3],[11-7]…..}. Hence the series is said to be an Arithmetic Sequence.

There are two Arithmetic Sequence formulas:-

• To find the nth term of an Arithmetic Sequences

• To find the sum of the first n terms of Arithmetic Sequences

Arithmetic Sequences Formula

an = a1 + (n-1)d

Where,

an = the nth term in the sequence

a1 = first term of the sequence

D = common difference between terms

Arithmetic Sequences Example

Let’s take an example for a better understanding 

• The sequence is 6,12,18,24,30 …. is an arithmetic sequence because every term is obtained by adding a constant number (6) to its previous term.

Here,

In the first term, a = 6

The common difference, d = 12 – 6 = 18 – 12 = 24 – 18 = … = 6

Thus, an arithmetic sequence can be written as a, a + d, a + 2d, a + 3d,… The verification for the above-given series formula can be written as:-

a, a + d, a + 2d, a + 3d, a + 4d, … = 6, 6 + 6, 6 + 2(6), 6 + 3(6), 6 + 4(6),… = 6, 12, 18, 24, 30,….

Given below are a few more examples of Arithmetic Sequences:

• 4,8,12,16,20, …

• 22,33,44,55,66, …

• π/2, π, 3π/2, 2π,…

• -3√2, -4√2, -5√2, -6√2, …

Terms Related to Arithmetic Sequence

Given below are the universally accepted terminology in Arithmetic Sequence used by one and all, a1, a2, a3, a4…….

First Term of Arithmetic Sequence

The first number in an Arithmetic Sequence is known as its First Term. It is usually represented by a₁ (or) a. For example, in the sequences 4, 8, 12, 16, … the first term is 4, i.e., a₁ = 4 (or) a = 4

Common Difference of Arithmetic Sequence

We have already seen that in an arithmetic sequence, each term, except the first term, is obtained by adding a fixed number to its previous term. Here, the “fixed number” is called the “common difference” and is denoted by ‘d’, and the formula for the common difference is d = aₙ – aₙ₋₁.

Nth Term of Arithmetic Sequence Formula

The nth term of an arithmetic sequence a₁, a₂, a₃, … is given by aₙ = a₁ + (n – 1) d. In an arithmetic sequence, it is also known as a general term. This directly follows from the understanding that the arithmetic sequence a₁, a₂, a₃, … = a₁, a₁ + d, a₁ + 2d, a₁ + 3d,… 

To find the sum of its first n terms, we calculate the sum of the Arithmetic Sequence Formula. Consider an arithmetic sequence where the first term is a₁ (or ‘a’), and the common difference is d. 

Sₙ denotes the sum of its first n terms.

• When the nth term is NOT known: 

Sₙ= n/2 [2a₁ + (n-1) d]

• When the nth term is known: 

Sₙ= n/2 [a₁ + aₙ]

Here are the formulas related to the arithmetic sequence.

• The common difference is d = a₂ – a₁.

• nth term, aₙ = a + (n – 1)d

• Sum of n terms, Sₙ = [n(a₁ + aₙ)]/2 (or) n/2 (2a + (n – 1)d)

Let’s take an example. Calculate the sum of the first 20 terms and calculate the 10th term for the given A.P. 4, 9, 11, 16, 21…. n.

For the sum of the first 20 terms,

Sn = n/2 (2a + (n−1)d)

Here, n = 20

           a = 4

           d = 5

Therefore, Sn = n/2 (2a+ (n−1)d)

S20 = 20/2 (2 x 4 + (20−1)5)

 = 1030

For the 10th term,

an = a1+ (n−1)d 

a10 = 4+ (10−1)5 

= 49

Conclusion

Understanding Arithmetic sequences and series formulas are integral to understanding patterns in number sequences. It helps in calculating unknown values when the pattern or the set of rules that a system follows is known. The formulas for the different series help in predictive analysis in other fields sloping to make better decisions by monitoring the outcome. Certain progressions follow a general rule or formula for evaluating the sum of numbers or the unknown term, thereby greatly reducing the stress and need for difficult and tedious calculations.