Arithmetic Progression (AP) is one of the most important and widely used concepts in mathematics. It appears in exam problems, real-life calculations, savings plans, installment payments, and number patterns. AP formulas help you find any term in a sequence or the total sum of selected terms. Once you know the first term, the common difference, and the position of the term, you can quickly calculate results without writing the entire sequence. These formulas save time, reduce errors, and are essential for students preparing for CUET, SSC, Banking, JEE, Railways, and school-level maths.
Formula for Calculating Arithmetic Progression (AP) - Overview
| Formula | Variables | When It Is Used |
|---|---|---|
| an = a + (n – 1)d | a = first term | To find the nth term of AP |
| Sn = n/2 [2a + (n – 1)d] | d = common difference | To find the sum of n terms |
| — | n = number of terms | Useful in exams & sequences |
What is Arithmetic Progression (AP) in Maths?
An Arithmetic Progression (AP) is a number sequence in which each term increases or decreases by a constant value. This constant value is called the common difference (d). For example, in the sequence 2, 4, 6, 8, …, the common difference is 2.
AP is one of the most commonly used sequences in mathematics. To find any term of an AP, we use the formula an = a + (n – 1)d, where you need only the first term, the common difference, and the term number. For calculating the total of the first n terms, we use Sn = n/2 [2a + (n – 1)d], which is much faster than adding each term manually.
Arithmetic Progression is used in competitive exams, banking interest patterns, saving plans, loan installments, and various real-life situations requiring uniform change.
Examples to Calculate Arithmetic Progression (AP)
Example 1: Find the 10th term of the AP 3, 7, 11, …
Step 1: a = 3, d = 4, n = 10
Step 2: an = a + (n – 1)d
= 3 + (10 – 1) × 4
= 3 + 36
= 39
So, the 10th term of the AP is 39.
Example 2: Find the sum of the first 20 terms of AP 5, 10, 15, …
Step 1: a = 5, d = 5, n = 20
Step 2: Sn = n/2 [2a + (n – 1)d]
= 20/2 [2×5 + (20 − 1)×5]
= 10 [10 + 95]
= 10 × 105
= 1050
So, the sum of the first 20 terms of the AP is 1050.
FAQs about Arithmetic Progression (AP) Formula
Q1. What is the common difference in AP?
It is the difference between any two consecutive terms. Example: In 7, 11, 15…, the common difference d = 4.
Q2. Can AP include negative numbers?
Yes, an AP can have positive, negative, or zero values. Example: –2, –5, –8…
Q3. Is every number sequence an AP?
No, only sequences where the difference between consecutive terms is constant are AP.
Q4. Why is AP important for competitive exams?
Because AP questions are frequently asked in CUET, SSC, Banking, JEE, Railways, and aptitude tests.
Q5. What is the sum of the first n natural numbers?
Using the AP formula with a = 1 and d = 1, the sum becomes n(n + 1)/2.
Q6. Who discovered Arithmetic Progression?
AP was used by ancient mathematicians, but Carl Friedrich Gauss is famously known for using AP concepts for fast calculations.