Last updated on July 7th, 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, AP series like 1, 6, 11, 16,... for instance, have 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.
An arithmetic sequence is one where the difference between any two successive terms is the same. E.g., in the arithmetic sequence given below, every term is obtained by adding 4, to its previous term.
To continue an arithmetic sequence, it is necessary to identify the common difference. Subtracting two consecutive terms helps us determine the common difference, which helps determine if the sequence is rising or declining. Next, add the common difference to the previous term to find the next term.
The formula for an arithmetic sequence is as follows:
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.
a1 = 2
a2 = 2 + 6
a3 = 2 + (2 × 6)
a4 = 2 + (3 × 6), and so on.
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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:
where
an= nth term,
a1 = first term,
and π 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 (π) is 4. Let’s substitute the equation to determine the seventh term a7:
a7 = 5 + (7 - 1) 4
= 5 + (6) 4
= 5 + 24
= 29
An arithmetic sequence's recursive formula is written as an=an-1+d, where an is the general or nth term, an-1 is the preceding term; π 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 a1 = 2 and an = an-1+3 for n > 1. It means that to calculate a new term, we have to add 3 to the previous term.
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:
Sn=n/2(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 π, the common difference π, and the number of terms π:
Sn=n/2 {2a+(n-1) d}
Where Sn is the sum of all the terms
Arithmetic sequences aren't just found in schoolbooks; they show up in a lot of real-life scenarios, too. Knowing how they work will help us make accurate decisions in a structured and mathematical way. This is useful for everything from saving money to building things. Let us take a look at some of the applications below:
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. Also, consistent monthly expenses, such as rent or utility bills, typically adhere to a predictable pattern. This pattern can be used to predict future expenses through arithmetic sequences.
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. Knowing these math patterns is important for making sure that structures like ramps, steps, and stacked walls are safe, symmetrical, and efficient.
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. This steady improvement makes it easier to track academic progress, set attainable goals, and estimate what the results will be in the future. Teachers and parents can use these patterns to help kids score better.
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. Designers and event planners also use these patterns to organize events effectively so that the largest number of people can fit at any given point in time.
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:
Find the 12th term in the arithmetic sequence: 5, 9, 13, 17, β¦
49
In the first step, we identify the common difference and the first term
Step 2: Use the formula given below to find out the nth term:
Step 3: Substitute the values into the formula:
a12=5+ 12-1 × 4
= 5+(11 × 4)
= 5 + 44 = 49
Therefore, the final answer will be 49.
Add up the first 10 numbers in this list: 2, 6, 10, 14,...
200
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 Sn=n/2 {2a+(n-1) d}, for the sum of n terms.
Step 3: Substitute the values:
S10=10/2{2×2+(10-1)×4}
=5(4+36)
=5×40
=200
Therefore, the final answer will be 200.
How many terms are there in this list: 7, 12, 17,..., 97?
19
First, list the numbers that are known.
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) ⇒ 90
=5(n-1)905
=n-118
=n18+1
=n19
Therefore, the sequence has 19 terms.
The 20th number in a sequence is 95, and the difference between them is 4. Find the first term.
19
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.
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.
550
Step 1: If the first and last terms are known, use the sum formula:
Sn=n/2(a+l)
Step 2: Substitute the values:
S10=10/2(10+100)
=5×110=550
The final answer will be 550.
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.
: He loves to play the quiz with kids through algebra to make kids love it.