A sequence is a collection of numbers that follow a pattern. An arithmetic sequence is a sequence of numbers where the differences between every two consecutive numbers are the same. 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 nth term of an arithmetic sequence
- The formula to find the sum of first n terms of an arithmetic sequence
Let us learn the definition of an arithmetic sequence and arithmetic sequence formulas along with derivations and a lot more examples.
What is an Arithmetic Sequence?
An arithmetic sequence 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. . Here is an arithmetic sequence example.
Arithmetic Sequences 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.
- The first term, a = 3
- The common difference, d = 6 - 3 = 9 - 6 = 12 - 9 = 15 - 12 = ... = 3
Thus, an arithmetic sequences 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, ...
Terms Related to Arithmetic Sequence
The terms of an arithmetic sequence is usually denoted by a₁, a₂, a₃, ... .We use the following terminology when we are dealing with arithmetic sequences.
First Term of Arithmetic Sequence
The first term of an arithmetic sequence, as its name suggests, is its first number. It is usually represented by a₁ (or) a. For example, in the sequence 5, 8, 11, 14, ... the first term is 5. i.e., a₁ = 6 (or) a = 6.
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. 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 nth term.
|Arithmetic sequence||First Term
|80, 75, 70, 65, 60, ...||80||-5||80 + (n - 1) (-5)
= -5n + 85
|π/2, π, 3π/2, 2π, ....||π/2||π/2||π/2 + (n - 1) (π/2)
-√2, -2√2, -3√2, -4√2, ...
|-√2||-√2||-√2 + (n - 1) (-√2)
= -√2 n
Arithmetic Sequence Recursive Formula
The above formula for finding the nth 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 nth term which is called the "recursive formula of an arithmetic sequence" and is used to find a term (aₙ) of the sequence when its previous term (aₙ₋₁) and 'd' are known. It says
aₙ = aₙ₋₁ + d
This formula just follows from the definition of the arithmetic sequence.
Example: Find a₂₁ of an arithmetic sequence if a₁₉ = -72 and d = 7.
By using the recursive formula,
a₂₀ = a₁₉ + d = -72 + 7 = -65
a₂₁ = a₂₀ + d = -65 + 7 = -58
Therefore, a₂₁ = -58.
Sum of Arithmetic sequence Formula
The sum of the arithmetic sequence formula is used to find the sum of its first n terms. Consider an arithmetic sequence 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 Sₙ. Then
- 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]
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?
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,
Sₙ = n/2 [2a+(n-1) d]
Sₙ = 5/2(2(200000) + (5 - 1)(25000))
= 5/2 (400000 +100000)
= 5/2 (500000)
She earns $1,250,000 in 5 years. We can this formula to be more helpful for larger values of 'n'.
Sum of Arithmetic Sequence Proof
Let us take an arithmetic sequence which has its first term to be a₁ and common difference to be d. Then the sum of the first 'n' terms of the sequence is given by
Sₙ = a₁ + (a₁ + d) + (a₁ + 2d) + … + aₙ ... (1)
Let us write the same sum from right to left (i.e., from the nth term to the first term).
Sₙ = aₙ + (aₙ – d) + (aₙ – 2d) + … + a₁ ... (2)
Adding (1) and (2), all terms with 'd' get canceled.
2Sₙ = (a₁ + aₙ) + (a₁ + aₙ) + (a₁ + aₙ) + … + (a₁ + aₙ)
2Sₙ = n (a₁ + aₙ)
Sₙ = [n(a₁ + aₙ)]/2
By substituting aₙ = a₁ + (n – 1)d into the last formula, we have
Sₙ = n/2 [a₁ + a₁ + (n – 1)d] (or)
Sₙ = n/2 [2a₁ + (n – 1)d]
Thus, we have derived both formulas for the sum of the arithmetic sequence.
List of Arithmetic Sequence Formulas
Here are the formulas related to arithmetic sequence.
- Common difference, 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)
Difference Between Arithmetic and Geometric Sequence
Here are the differences between arithmetic and geometric sequence:
|Arithmetic sequences||Geometric sequences|
|In this, the differences between every two consecutive numbers are the same.||In this, the ratios of every two consecutive numbers are the same.|
|It is identified by the first term (a) and the common difference (d).||It is identified by the first term (a) and the common ratio (r).|
|There is a linear relationship between the terms.||There is an exponential relationship between the terms.|
Important Notes Related to 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 nth term of an arithmetic sequence is aₙ= a₁ + (n−1)d.
- The sum of the first n terms of an arithmetic sequence is Sₙ = n/2[2a₁ + (n − 1)d].
- The common difference of arithmetic sequences can be either positive or negative or zero.
Topics Related to Arithmetic Sequence
Solved Examples on Arithmetic Sequence
Example 1: Find the nth term of the arithmetic sequence -5, -7/2, -2, ....
The given sequence is -5, -7/2, -2, ...
Here, the first term is a = -5, and the common difference is, d = -(7/2) - (-5) = -2 - (-7/2) = ... = 3/2.
The nth term of an arithmetic sequence is given by
aₙ = a + (n - 1)d
aₙ = -5 +(n - 1) (3/2)
= -5+ (3/2)n - 3/2
= 3n/2 - 13/2
Answer: The nth term of the given arithmetic sequence is, aₙ = 3n/2 - 13/2.
Example 2: Which term of the arithmetic sequence -3, -8, -13, -18,... is -248?
The given arithmetic sequence is -3, -8, -13, -18,...
The first term is, a = -3
The common difference is, d = -8 - (-3) = -13 - (-8) = ... = -5.
It is given that nth term is, aₙ = -248.
Substitute all these values in the nthl term of an arithmetic sequence formula,
aₙ = a + (n - 1)d
-248 = -3 + (-5)(n - 1)
-248 = -3 -5n + 5
-248 = 2 - 5n
-250 = -5n
n = 50
Answer: -248 is the 50th term of the given sequence.
FAQs on Arithmetic sequence
What is 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 Find 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
- 8-3 = 5
- 13-8 = 5
- 18-13 = 5 and so on.
What is the nth term of an Arithmetic Sequence Formula?
The nth term of arithmetic sequences is given by aₙ = a + (n – 1) × d. Here 'a' represents the first term and 'd' represents the common difference.
What is the Sum of an Arithmetic Sequence Formula?
The sum of the first n terms of an arithmetic sequence with the first term 'a' and common difference 'd' is denoted by Sₙ and we have two formulas to find it.
- Sₙ = n/2[2a + (n - 1)d]
- Sₙ = n/2[a + aₙ].
What is the Formula to Find the Common Difference of 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 = aₙ - aₙ₋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, aₙ or Sₙ might have been given in the problem. We will just substitute the given values in the formulas of aₙ or Sₙ 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) aₙ (or) Sₙ might be given in the problem. We will just substitute the given values in the formulas of aₙ or Sₙ 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 nth Term in Arithmetic Sequence?
Here are the steps for finding the nth 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 aₙ = 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 Sₙ= n/2(2a + (n - 1)d)