Arithmetic Sequence
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Last updated on December 12, 2025

An arithmetic sequence or arithmetic progression is a set of numbers where the common difference between any two consecutive terms is constant. For example, an AP series like 1, 6, 11, 16,... has a common difference of five. There are formulas to help us determine the nth term and the sum of the first n terms in an arithmetic sequence. In this article, we will discuss arithmetic sequences in detail.

What is an Arithmetic Sequence?

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An arithmetic sequence is a list of numbers where the difference between any two successive terms is the same. This constant difference is called common difference.

For example, in the arithmetic sequence given below, every term is obtained by adding 4 to its previous term.

4, 8, 16, 20, . . . 

Here, the common difference, denoted by (d), is 4.

How to Continue an Arithmetic Sequence?

To continue an arithmetic sequence, follow the given steps.

Step 1: Look at two consecutive terms. Pick any pair neighboring each other.

Step 2: Find the common difference (d).

Step 3: Check whether the sequence is rising \((d > 0)\), steady \((d = 0)\) or falling \((d < 0)\).

Step 4: Find the next term by adding the common difference to the last term.

Step 5: Repeat the addition to continue the sequence.

  • To find a term at any position in the sequence, use the nth term formula.

What is the Formula of Arithmetic Sequence?

The formula for an arithmetic sequence is as follows:


\(a_n = a_1 + (n - 1) \times d \)


Here,
 

  • an is the general or nth term
     
  • a1 stands for the first term
     
  • n is the position of the term, and 
     
  • d is the common difference.

To understand the formula better, let’s take an example:
 

2, 8, 14, 20, 26, ....


In the above sequence, d is 6. 

  • \(a_1 = 2 \)
     
  • \(a_2 = 2 + 6 \)
     
  • \(a_3 = 2 + (2 × 6) \)
     
  • \(a_4 = 2 + (3 × 6) \), and so on.
    .
    .
    .
    .
  • \(an = a_1 + (n - 1) × d \)

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What is the nth Term of Arithmetic Sequence?

The value at a specific position in an arithmetic sequence is represented by the nth term. The following formula can be used to find it:

          \(   a_n = a_1 + (n - 1) × d     \)

Where, 
 

  • an = nth term, 
     
  • a1 = first term, 
     
  • and 𝑑 is the common difference between the terms.

For example, a sequence like 5, 9, 13, 17,..... each number rises by 4. Hence, the first term a1 is 5, and the common difference (d) is 4. Let’s substitute the equation to determine the seventh term, a7:

\(a_7 = 5 + (7 - 1) × 4 \)

     \(  = 5 + (6 × 4) \)

     \( = 5 + 24\)

     \(= 29 \)

What is the Recursive Formula of Arithmetic Sequence

An arithmetic sequence's recursive formula is written as:

\(a_n = a_{n-1} + d\)

Where,

  • an is the general or nth term,
     
  • an-1 is the preceding term;
     
  • 𝑑 is the common difference between terms.
     

The initial term (a1) must be utilized to apply the recursive formula. In the sequence 2, 5, 8, 11, and so on, for instance, the first term is 2, and the common difference is 3.
 

The recursive formula then is \(a_1 = 2 \) and \(a_n = a_{n-1} + 3\) for n > 1. It means that to calculate a new term, we have to add 3 to the previous term.

What is the Sum of Arithmetic Sequence

The sum of an arithmetic sequence is resulted by adding all the terms of the sequence. The formula to calculate the sum of an arithmetic sequence is given below:
 

                           \(S_n = \frac{n}{2} \times (a + l) \)

Where,
 

  • Sn is the sum of the sequence up to the nth term
     
  • a is the first term
     
  • l is the last term, and 
     
  • n is the number of terms.

Alternatively, we can also use the below-mentioned formula if we know the first term 𝑎, the common difference 𝑑, and the number of terms 𝑛:

                                                
    \(S_n = \frac{n}{2} \times [2a + (n - 1)d] \)

Where,
 

  • Sn is the sum of all the terms
     
  • a is the first term,
     
  • d is the common difference, and 
     
  • n is the number of terms.
     
  • 2a is 2 multiplied by the first term (when a is the first term of the sequence)

Difference Between Arithmetic Sequence and Geometric Sequence

Here are the differences between an arithmetic sequence and a geometric sequence:

Arithmetic Sequence Geometric Sequence The difference between any two consecutive terms is constant. The ratio between any two consecutive terms is constant. It is defined by its first term (a) and common difference (d). It is defined by its first term (a) and standard ratio (r). Its terms increase linearly. Its terms grow exponentially.

Arithmetic Series

The sum of an arithmetic sequence helps us find the total of the first n terms of the sequence. This total is called an arithmetic series. If the first term is a1 (or a) and the common difference is d, the sum of the first n terms (Sn) can be found using one of these formulas.

When the nth term is not known,
 

\(S_n = \frac{n}{2} \left[ 2a_1 + (n - 1)d \right] \)

When the nth term is known,

\(S_n = \frac{n}{2}(a_1 + a_n) \)

For example,
Natalie earns 200,000 dollars in her first year, and her salary increases by 25,000 dollars every year. How much does she earn in total during her first 5 years?

First year salary, a = 200,000

Yearly increase, d = 25,000

Number of years, n = 5

By using the formula,

\(S_n = \frac{n}{2}\left[2a + (n-1)d\right] \)

\(S_5 = \frac{5}{2}\left[2(200000) + 4(25000)\right] \)

\(= \frac{5}{2}(400000 + 100000) \)

\(= \frac{5}{2}(500000) = 1,250,000 \)

So, Natalie earns 1,250,000 in her first five years. This method is beneficial when calculating sums for large values of n.

Important Notes on Arithmetic Sequence

In an arithmetic sequence, the difference between each pair of consecutive terms remains constant.

  • This constant value, called the common difference, is given by:

    \(d = a_2 - a_1 = a_3 - a_2 = \ldots \)

  • The nth term of an arithmetic sequence is found using:

    \(a_n = a_1 + (n - 1)d \)

  • The sum of the first n terms is calculated using:

    \(s_n = \frac{n}{2}\left[2a_1 + (n - 1)d\right] \)

  • The common difference can be positive, negative, or even zero, depending on how the sequence progresses.

Tips and Tricks to Master Arithmetic Sequence

Understanding arithmetic sequences is easier when you know the proper techniques. Use these helpful tricks to find common differences, calculate terms, and confidently tackle sequence-based math problems with ease.

  • Always identify the pattern first, as every term changes by the same difference.
     
  • Use subtraction to quickly find the common difference.
     
  • Use the nth term formula to reach any term instead of listing them out one by one.
     
  • Check for negative and positive signs to identify falling and rising sequences, respectively.
     
  • To find the total sum, use \(S_n=\frac{n}{2}(2a+(n-1)d)\). This is a shortcut to quickly see the total.
     
  • Children should always look for the pattern first, as every term changes by the same difference.
     
  • Teachers can encourage students to verify answers using formulas.
     
  • Parents can encourage and motivate children during practice.

Common Mistakes and How to Avoid Them in Arithmetic Sequence

It is not uncommon for students to make mistakes while working on an arithmetic sequence. This section talks about some of those mistakes and the solutions to avoid them:

Real-Life Applications of Arithmetic Sequence

Arithmetic sequences show up in a lot of real-life scenarios. Knowing how they work will help us make accurate decisions in a structured and mathematical way. 

  • Monthly Savings and Budgeting: When an individual sets aside $2000 every month, the savings form an arithmetic sequence: $2000, $4000, $6000, and so on. This pattern aids in financial planning by enabling individuals to forecast their savings after a designated number of months.
  • Building and designing stairs: Structures with evenly increasing levels, like stairs, often follow a pattern. E.g., each step on a ladder might be 6 inches higher than the one before it. Architects and builders can estimate the total rise, the number of steps, and the materials required more easily with this steady rise. 
  • Plans for mobile data or subscriptions: Some mobile plans or subscription services offer perks that keep increasing over time. As an example, a person might get 1GB of internet data in the first month, 2GB in the second, 3GB in the third, and so on. Users can then choose the right plan based on their knowledge about future data limits or service benefits.
  • Tracking student performance: Suppose a student improves their score on each test by the same number of points, say 5 points each time. This is called an arithmetic sequence. For instance, their scores could be 60, 65, 70, 75, and so on. 
  • Seating arrangements in theaters and halls: Many theaters and stadiums are built in such a way that each row has more seats than the one before it. For example, each row might have two more seats than the row before it. This forms a pattern and an arithmetic sequence, making it easier for engineers to construct these halls or stadiums. 

Problem 1

Find the 12th term in the arithmetic sequence: 5, 9, 13, 17, …

Okay, lets begin

49

Explanation

In the first step, we identify the common difference and the first term

  • First term \(a = 5\), and 
     
  • Common difference \(d = 9 - 5 = 4\)

Step 2: Use the formula find out the nth term.

Step 3: Substitute the values into the formula:

             

\(a_{12} = 5 + (12 - 1) × 4\)

             \( = 5 + (11 × 4)\)

            \(  = 5 + 44 = 49\)

Therefore, the final answer will be 49.

Well explained 👍

Problem 2

Add up the first 10 numbers in this list: 2, 6, 10, 14,...

Okay, lets begin

200

Explanation

Step 1: List the known parameters
 

  • The first term is \(a = 2\)
     
  • Common difference, \(d = 6 – 2 = 4\)
     
  • Number of terms \(n = 10\)

Step 2: Use the formula \(S_n = \frac{n}{2} \times [2a + (n - 1)d] \), for the sum of n terms.

Step 3: Substitute the values:
 

              \(S_{10} = \frac{10}{2} \times [2 \times 2 + (10 - 1) \times 4] \)

                   \(  = 5 × (4 + 36)\)

                   \(  = 5 × 40\)

                  \(   = 200\)

Therefore, the final answer will be 200.

Well explained 👍

Problem 3

How many terms are there in this list: 7, 12, 17,..., 97?

Okay, lets begin

19

Explanation

First, list the numbers that are known.

  • The first term is \(a = 7\)
     
  • Difference in common \(d = 12 – 7 = 5\)
     
  • Last term \(l = 97\)

Step 2: Use the nth term formula and solve for n:

            \(  l = a + (n - 1) d ⇒ 97\)

           \(     = 7 + (n - 1) × 5\)

Step 3: Solve the equation:
 

        \(97 - 7 = 5 + (n - 1) \)

      \(  ⇒ 5 + (n - 1) = 90\)

     

\(  ⇒ 5n = 90 + 5\)

      \(  ⇒ 5n = 95\)

        ⇒ n = 19

Therefore, the sequence has 19 terms.

Well explained 👍

Problem 4

The 20th number in a sequence is 95, and the difference between them is 4. Find the first term.

Okay, lets begin

19

Explanation

First, use the following method to find the nth term:             

Step 2: Substitute the values:

\(95 = a + (20 - 1) × 495\)
\(95 = a + 76\)

Step 3: Solve for the value of a:
 

             \(a = 95 -76 = 19\)


The first term will be 19.

Well explained 👍

Problem 5

There are 10 terms in an arithmetic sequence, with 10 being the first term and 100 being the last. Find the sum of these terms.

Okay, lets begin

550

Explanation

Step 1: If the first and last terms are known, use the sum formula:

  \(S_n = \frac{n}{2} \times (a + l) \)

Step 2: Substitute the values:
 

\(S_{10} = \frac{10}{2} \times (10 + 100) \)

\(= 5 × 110 = 550\)


The final answer will be 550.
 

Well explained 👍

FAQs in Arithmetic Sequence

1. Define an arithmetic sequence.

An arithmetic sequence is a collection of numbers in which each term is generated by adding the same constant value, known as the common difference, to the previous term. The sequence either expands or contracts. For instance, 2, 5, 8, 11 represents an expanding sequence, while 20, 15, 10, 5 indicates contraction.

2.What defines an arithmetic sequence?

Check whether the difference between any two consecutive terms is constant to identify an arithmetic sequence. Determine if the difference stays the same throughout the sequence. If there are any variations, the sequence does not follow arithmetic. 

3.The nth term of an arithmetic sequence has what formula?

 An arithmetic sequence's nth term formula, \(aₙ = a₁ + (n − 1) × d\), allows one to find any term without listing all prior terms. Here, the first term is 𝑎1; the common difference is 𝑑, and the position is 𝑛. The formula makes calculations easy, especially for a long sequence.

4.Does a negative common difference characterize an arithmetic sequence?

Arithmetic sequences can indeed have negative common differences. Thus, the sequence is declining, and every term is smaller than the one before it. The common difference among 30, 25, 20, and 15 is –5. A negative common difference denotes a consistent decline. 

5.What happens if you miss a term? Is it still arithmetic?

If a term is absent, we cannot quickly verify whether the sequence is arithmetic or not. We have to see if the known terms maintain a constant difference. Missing data could make it uncertain if the trend continues. Many times, proper identification calls for the replacement of a missing value, depending on the arithmetic pattern.

6.How can I help my child identify the common difference in an arithmetic sequence?

To help your child identify the common difference in an arithmetic sequence, encourage them to look for the pattern of change between consecutive terms. Ask them to subtract any term from the next one if the result is always the same, that number is the common difference. For example, in the sequence 5, 9, 13, 17…, each term increases by 4, so the common difference is +4.

7.How do arithmetic sequences differ from geometric sequences?

Arithmetic sequences and geometric sequences differ in how their terms progress. In an arithmetic sequence, each term is obtained by adding or subtracting a fixed number called the common difference. For example, 3, 6, 9, 12… increases by 3 each time. In contrast, a geometric sequence is formed by multiplying or dividing each term by a constant called the common ratio.

8.Can arithmetic sequences have fractions or decimals as terms?

Yes, arithmetic sequences can definitely include fractions or decimals as terms. What matters is that the difference between consecutive terms remains constant, regardless of the type of number

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.