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Last updated on September 16, 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.
An arithmetic sequence is one where the difference between any two successive terms is the same. Example, 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:
an = a1 + (n - 1) × d
Here,
To understand the formula better, let’s take an example:
2, 8, 14, 20, 26, ....
In the above sequence, d is 6.
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:
an = a1 + (n - 1) × d
where
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:
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
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
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
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:
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
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)
⇒ 5 + (n - 1) = 90
⇒ 5n = 90 + 5
⇒ 5n = 95
⇒ n = 19
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.