Arithmetic Sequence

The arithmetic sequence is the sequence where the common difference remains constant between any two successive terms. Let us recall what is a sequence. A sequence is a collection of numbers that follow a pattern. For example, the sequence 1, 6, 11, 16, … is an arithmetic sequence because there is a pattern where each number is obtained by adding 5 to its previous term. We have two arithmetic sequence formulas.

• The formula for finding n^th term of an arithmetic sequence
• The formula to find the sum of first n terms of an arithmetic sequence

If we want to find any term in the arithmetic sequence then we can use the arithmetic sequence formula. Let us learn the definition of an arithmetic sequence and arithmetic sequence formulas along with derivations and a lot more examples for a better understanding.

An arithmetic sequence is defined in two ways. It is a "sequence where the differences between every two successive terms are the same" (or) In an arithmetic sequence, "every term is obtained by adding a fixed number (positive or negative or zero) to its previous term". The following is an arithmetic sequence as every term is obtained by adding a fixed number 4 to its previous term.

Arithmetic Sequence Example

Consider the sequence 3, 6, 9, 12, 15, .... is an arithmetic sequence because every term is obtained by adding a constant number (3) to its previous term.

Here,

• The first term, a = 3
• The common difference, d = 6 - 3 = 9 - 6 = 12 - 9 = 15 - 12 = ... = 3

Thus, an arithmetic sequence can be written as a, a + d, a + 2d, a + 3d, .... Let us verify this pattern for the above example.

a, a + d, a + 2d, a + 3d, a + 4d, ... = 3, 3 + 3, 3 + 2(3), 3 + 3(3), 3 + 4(3),... = 3, 6, 9, 12,15,....

A few more examples of an arithmetic sequence are:

• 5, 8, 11, 14, ...
• 80, 75, 70, 65, 60, ...
• π/2, π, 3π/2, 2π, ....
• -√2, -2√2, -3√2, -4√2, ...

The first term of an arithmetic sequence is a, its common difference is d, n is the number of terms. The general form of the AP is a, a+d, a+2d, a+3d,......up to n terms. We have different formulas associated with an arithmetic sequence used to calculate the n^th term, the sum of n terms of an AP, or the common difference of a given arithmetic sequence.

The arithmetic sequence formula is given as,

• N^th Term: a_n = a + (n-1)d
• S_n = (n/2) [2a + (n - 1)d]
• d = a_n - a_n-1

The n^th term of an arithmetic sequence a₁, a₂, a₃, ... is given by a_n = a₁ + (n - 1) d. This is also known as the general term of the arithmetic sequence. This directly follows from the understanding that the arithmetic sequence a₁, a₂, a₃, ... = a₁, a₁ + d, a₁ + 2d, a₁ + 3d,... The following table shows some arithmetic sequences along with the first term, the common difference, and the n^th term.

Arithmetic Sequence Recursive Formula

The above formula for finding the n^th term of an arithmetic sequence is used to find any term of the sequence when the values of 'a₁' and 'd' are known. There is another formula to find the n^th term which is called the "recursive formula of an arithmetic sequence" and is used to find a term (a_n) of the sequence when its previous term (a_n-1) and 'd' are known. It says

a_n = a_n-1 + d

This formula just follows the definition of the arithmetic sequence.

Example: Find a₂₁ of an arithmetic sequence if a₁₉ = -72 and d = 7.

Solution:

By using the recursive formula,

a₂₀ = a₁₉ + d = -72 + 7 = -65

a₂₁ = a₂₀ + d = -65 + 7 = -58

Therefore, a₂₁ = -58.

The sum of the arithmetic sequence formula is used to find the sum of its first n terms. Note that the sum of terms of an arithmetic sequence is known as arithmetic series. Consider an arithmetic series in which the first term is a₁ (or 'a') and the common difference is d. The sum of its first n terms is denoted by S_n. Then

• When the n^th term is NOT known: S_n= n/2 [2a₁ + (n-1) d]
• When the n^th term is known: S_n = n/2 [a₁ + a_n]

Example

Ms. Natalie earns $200,000 per annum and her salary increases by $25,000 per annum. Then how much does she earn at the end of the first 5 years?

Solution:

The amount earned by Ms. Natalie for the first year is, a = 2,00,000. The increment per annum is, d = 25,000. We have to calculate her earnings in the first 5 years. Hence n = 5. Substituting these values in the sum sum of arithmetic sequence formula,

S_n= n/2 [2a₁ + (n-1) d]

⇒ S_n = 5/2(2(200000) + (5 - 1)(25000))

= 5/2 (400000 +100000)

= 5/2 (500000)

= 1250000

She earns $1,250,000 in 5 years. We can use this formula to be more helpful for larger values of 'n'.

Sum of Arithmetic Sequence

Let us take an arithmetic sequence that has its first term to be a₁ and the common difference to be d. Then the sum of the first 'n' terms of the sequence is given by

S_n = a₁ + (a₁ + d) + (a₁ + 2d) + … + a_n ... (1)

Let us write the same sum from right to left (i.e., from the n^th term to the first term).

S_n = a_n + (a_n – d) + (a_n – 2d) + … + a₁ ... (2)

Adding (1) and (2), all terms with 'd' get canceled.

2S_n = (a₁ + a_n) + (a₁ + a_n) + (a₁ + a_n) + … + (a₁ + a_n)

2S_n = n (a₁ + a_n)

S_n = [n(a₁ + a_n)]/2

By substituting a_n = a₁ + (n – 1)d into the last formula, we have

S_n = n/2 [a₁ + a₁ + (n – 1)d] (or)

S_n = n/2 [2a₁ + (n – 1)d]

Thus, we have derived both formulas for the sum of the arithmetic sequence.

Here are the differences between arithmetic and geometric sequence:

Important Notes on Arithmetic Sequence:

• In arithmetic sequences, the difference between every two successive numbers is the same.
• The common difference of an arithmetic sequence a₁, a₂, a₃, ... is, d = a₂ - a₁ = a₃ - a₂ = ...
• The n^th term of an arithmetic sequence is a_n = a₁ + (n−1)d.
• The sum of the first n terms of an arithmetic sequence is S_n = n/2[2a₁ + (n − 1)d].
• The common difference between arithmetic sequences can be either positive or negative or zero.

☛ Related Topics:

• Sequence Calculator
• Series Calculator
• Arithmetic Sequence Calculator
• Geometric Sequence Calculator

What is an Arithmetic Sequence in Algebra?

An arithmetic sequence in algebra is a sequence of numbers where the difference between every two consecutive terms is the same. Generally, the arithmetic sequence is written as a, a+d, a+2d, a+3d, ..., where a is the first term and d is the common difference.

What are Arithmetic Sequence Formulas?

Here are the formulas related to an arithmetic sequence where a₁ (or a) is the first term and d is a common difference:

• The common difference, d = a_n - a_n-1.
• n^th term of sequence is, a_n = a + (n - 1)d
• Sum of n terms of sequence is , S_n = [n(a₁ + a_n)]/2 (or) n/2 (2a + (n - 1)d)

What is the Definition of an Arithmetic Sequence?

A sequence of numbers in which every term (except the first term) is obtained by adding a constant number to the previous term is called an arithmetic sequence. For example, 1, 3, 5, 7, ... is an arithmetic sequence as every term is obtained by adding 2 (a constant number) to its previous term.

How to Identify An Arithmetic Sequence?

If the difference between every two consecutive terms of a sequence is the same then it is an arithmetic sequence. For example, 3, 8, 13, 18 ... is arithmetic because the consecutive terms have a fixed difference.

• 8-3 = 5
• 13-8 = 5
• 18-13 = 5 and so on.

What is the n^th term of an Arithmetic Sequence?

The n^th term of arithmetic sequences is given by a_n = a + (n – 1) × d. Here 'a' represents the first term and 'd' represents the common difference.

What is an Arithmetic Series?

An arithmetic series is a sum of an arithmetic sequence where each term is obtained by adding a fixed number to each previous term.

What is the Arithmetic Series Formula?

The sum of the first n terms of an arithmetic sequence (arithmetic series) with the first term 'a' and common difference 'd' is denoted by Sₙ and we have two formulas to find it.

• S_n = n/2[2a + (n - 1)d]
• S_n = n/2[a + a_n].

What is the Formula to Find the Common Difference in Arithmetic sequence?

The common difference of an arithmetic sequence, as its name suggests, is the difference between every two of its successive (or consecutive) terms. The formula for finding the common difference of an arithmetic sequence is, d = a_n - a_n-1.

How to Find n in Arithmetic Sequence?

When we have to find the number of terms (n) in arithmetic sequences, some of the information about a, d, a_n or S_n might have been given in the problem. We will just substitute the given values in the formulas of a_n or S_n and solve it for n.

How To Find the First Term in Arithmetic sequence?

The first term of an arithmetic sequence is the number that occurs in the first position from the left. It is denoted by 'a'. If 'a' is NOT given in the problem, then some information about d (or) a_n (or) S_n might be given in the problem. We will just substitute the given values in the formulas of a_n or S_n and solve it for 'a'.

What is the Difference Between Arithmetic Sequence and Arithmetic Series?

An arithmetic sequence is a collection of numbers in which all the differences between every two consecutive numbers are equal to a constant whereas an arithmetic series is the sum of a few or more terms of an arithmetic sequence.

What are the Types of Sequences?

There are mainly 3 types of sequences in math. They are:

• Arithmetic sequence
• Geometric sequence
• Harmonic sequence

What are the Applications of Arithmetic Sequence?

Here are some applications: the salary of a person which is increased by a constant amount by each year, the rent of a taxi which charges per mile, the number of fishes in a pond that increase by a constant number each month, etc.

How to Find the n^th Term in Arithmetic Sequence?

Here are the steps for finding the n^th term of arithmetic sequences:

• Identify its first term, a
• Common difference, d
• Identify which term you want. i.e., n
• Substitute all these into the formula a_n = a + (n – 1) × d.

How to Find the Sum of n Terms of Arithmetic Sequence?

To find the sum of the first n terms of arithmetic sequences,

• Identify its first term (a)
• Common difference (d)
• Identify which term you want (n)
• Substitute all these into the formula S_n= (n/2)(2a + (n - 1)d)