Arithmetic Sequence

Arithmetic Sequence is a type of sequence out of all sequences where each term of the sequence is related to the previous term of the sequence by a linear relation. A sequence is a collection of objects where all the terms follow an order or pattern by which the whole sequence can be identified. In the case of an Arithmetic Sequence, each term can be found by adding a constant to the preceding term of the Arithmetic Sequence, this constant sets the Arithmetic Sequence apart from the other sequences.

In this article, we will explore the concept of Arithmetic Sequence and various formulas associated with it. We will also learn about the various properties of Arithmetic Sequences.

In the arithmetic sequence, the absolute difference between one term and the next term is constant. In other words, a sequence a₁, a₂, . . ., a_n is called an arithmetic sequence or arithmetic progression if a_n+1 – a_n = d where d is constant and it is the common difference of the sequence.

Let’s make a general example of an Arithmetic Progression as follows:

a, a+d, a+2d, a+3d, . . .

Where 

• a is the first term of Arithmetic Sequence
• d is a common difference between any two consecutive terms of Arithmetic Sequence

Arithmetic Sequence Definition

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference.

Here are some examples of arithmetic sequences,

Example 1: Sequence of even number having difference 4 i.e., 2, 6, 10, 14, . . . ,

Here in the above example, the first term of the sequence is a₁=2 and the common difference is 4 = 6 -2.

Example 2: -3, 0, 3, 6, 9, 12, . . .

In this sequence, the common difference is 3. Each term is obtained by adding 3 to the preceding term.

Example 3: 100, 90, 80, 70, 60, …

In this sequence, the common difference is -10. Each term is obtained by subtracting 10 from the preceding term.

As we already discussed that the arithmetic sequence is a series of numbers where each number is calculated by adding a constant in the previous term. There are various formulas that are used in the Arithmetic sequence that are,

• N^th Term of Arithmetic Sequence
• Recursive Formula for Arithmetic Sequence
• Sum of terms in Arithmetic Sequence

Let’s consider an example of Arithmetic Sequence 6, 16, 26, 36, 46, 56, 66, . . .. 

As we can see each term of this example can be represented in a similar form. Thus, the n^th term can be found easily by adding one less than n multiple of 10 to the first term of the sequence i.e., 6. 

Thus, the n^th term of the given example can be generalized as 6 + (n-1)×d.

In general, this is the standard explicit formula of an arithmetic sequence whose first term is, A, end, and the common difference is D is given as follows:

a_n = a + (n-1)×d

Read more: How to Find the Nth term of Arithmetic Sequence?

Recursive Formula for Arithmetic Sequence

The n^th term of an Arithmetic Sequence can be defined recursively as the next term can always be obtained by adding a common difference to the preceding term, the following derivation can be used to illustrate the same thing.

As we know, n^th term of the Arithmetic Sequence is given by 

a_n = a + (n-1)×d

thus, (n-1)^th term can be given by 

a_n-1 =  a + (n-1-1)×d

a_n-1 = a + (n-2)×d

Thus,  

a_n = a + (n-1)×d

⇒ a_n = a+(n-1-1+1)×d

⇒ a_n = a + (n-2+1)×d 

⇒ a_n = a + (n-2)×d + d

⇒ a_n = a_n-1 + d

Sum of terms in Arithmetic Sequence

Let’s sequence is given as a, a+d, a+2d, a+3d, ….. a+(n-1)d.

S_n = (n/2)(a + l) 

where,

• a is the first term
• l is the last term of the series and 
• n is the number of terms in the series

Replacing the last term l by the n^th term = a + (n – 1)d, we get

S_n = (n/2)(a + a + (n – 1)d)

OR

S_n = (n/2)(2a + (n – 1) x d)

The sum of terms of an Arithmetic Sequence is called Arithmetic Series. We use the sum of n terms formula of the arithmetic sequence to find the sum of the arithmetic series.

Suppose the first term of the arithmetic series is a and the common difference is d then the sum of the n term of this arithmetic series is given using the formula,

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

If the common difference of the arithmetic series is not given but the nth term of the series is given (say l). Then its sum is calculated as,

S_n= n/2 [a + l]

We can understand this with the help of the example,

Example: A tree fruits five apples in the first year and at each successive year it has 2 more apples than the last year find the total apple the tree bears at the end of six years.

Solution:

Apple bear by tree in first year (a) = 5

Yearly Increase in apple bear by the tree (d) = 2

Time Period = 6 years

Total apple at the end of five years in the tree.

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

⇒ S_n = 6/2(2(5) + (6 – 1)(2))

⇒ S_n = 6/2 (10 + 10)

⇒ S_n = 3(20)

⇒ S_n = 60

Thus, the apple in the tree at the end of six-year is 60 apples

There are some properties of Arithmetic Sequence, some of which are as follows:

• If a constant is added or subtracted to each term of an Arithmetic Sequence then the resulting sequence is also an Arithmetic Sequence.
• If each term of an Arithmetic Sequence is multiplied or divided (not by 0) by a constant number. Then the resulting sequence is also an Arithmetic Sequence.
• For any three consecutive terms of an Arithmetic Sequence sum of the first and last term is always twice the middle term.
• We can observe a symmetry about the mean in the arithmetic sequence.
• An arithmetic sequence can be extended to infinity by adding a common difference to the last term.

The key differences between Arithmetic Sequence and Geometric Sequence are as follows:

Learn More: Difference between an Arithmetic Sequence and a Geometric Sequence

People Also Read:

• Sum of First n Terms
• Geometric Progression

Problem 1: Write the first three terms in each of the following sequences defined by 

• A_n=5n+2(n-1)
• A_n=2n +4(n-2)  

Solution :    

For A_n=5n+2(n-1)

Put n=1, we get 

a₁=5.1 + 2(1-1) = 5+ 0 =5

Put n=2, we get 

a₂=5.2+2(2-1)  =10+2 =12 

Put n=3, we get 

a₃=5.3 + 2(3-1) =15 + 4 =19

So first three terms are 5,12, 19.

For A_n=2n +4(n-2)  

 Put n=1, we get 

a₁=2.1+4(1-2) =2-4 = -2

Put n=2, we get 

a₂ = 2.2+4(2-2) =4+ 0 =4

Put n=3,we get 

a₃= 2.3 + 4(3-2) =6+4 =10

So the first three terms are -2, 4, 10.

Problem 2: Find the 20^th Term of the given expression A_n=(n-1)(2-n)(3+n).

Solution:

For A_n=(n-1)(2-n)(3+n)

Put n=20 in given  expression,

a₂₀ =(20-1)(2-20)(20+3) 

⇒ a₂₀ = 19×(-18)×(23) 

⇒ a₂₀ = -7886.

Problem 3: Find the sum of all natural numbers lying between 100 and 1000 (inclusive of both 100 and 1000) which are multiples of 5.

Solution:

Solve: first term to be 100 and last terms is 1000 and common difference is 5.

So our formula is S_n=(n/2)[2a+(n-1)×d] .

using a_n = a₁ + (n-1)d

⇒ 1000 = 100 + (n – 1)5

⇒ 900 =  (n – 1)5

⇒ 180 = n – 1

⇒ n = 181

Thus, there are 181 such number. Now for sum of all the 181 terms of sequence can be calculated as follows:

S₁₈₁ = (181/2)[2·100 +(181-1)×5].

⇒ S₁₈₁ = (181/2)[200+180×5]

⇒ S₁₈₁ = (181/2)×1100

⇒ S₁₈₁ = 181×550 = 99,550

1. Given the arithmetic sequence 2, 6, 10, 14,…… find the common difference and the 15th term.

2. If the first term of an arithmetic sequence is 10 and the common difference is 3, find the nth term of the sequence.

3. In an arithmetic sequence, the 7th term is 33 and the 15th term is 57. Find the common difference and the first term of the sequence.

4. An arithmetic sequence has a first term of 7 and a common difference of 4. Find the sum of the first 10 terms.

5. The 12th term of an arithmetic sequence is 30 and the 20th term is 50. Find the common difference and the first term of the sequence

Define Arithmetic Sequence.

Arithmetic Sequence is defined as the sequence where each term of the sequence can be calculated by adding a constant in the preceding term of the same sequence.

What is the Common Difference of Arithmetic Sequence?

The difference between two consecutive terms of an Arithmetic Sequence is called the common difference of an  Arithmetic Sequence.

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

The nth term of an arithmetic sequence is given by the following formula:

a_n = a+(n-1)×d

where,

• a is the first term of the sequence,
• d is the common difference of the sequence,
• n is the number of terms in the sequence,

How to find the Sum of the First n terms of an Arithmetic Sequence?

We can use the following formulas to find the sum of n terms of an arithmetic sequence.

S_n = (n/2)(a + l)

S_n = (n/2)(2a + (n – 1) x d)

where,

• a is the first term of the sequence,
• d is the common difference of the sequence,
• l is the last term of the sequence, and 
• n is the number of terms in the sequence,

What is the Difference between an Arithmetic Sequence and a Geometric Sequence?

In Arithmetic Sequence, the difference between any two consecutive terms is constant. Whereas in a geometric sequence, the ratio of any two consecutive terms is constant.