The nth term formula of an Arithmetic Progression (AP) helps you find any term in a sequence without listing all previous terms. Whether you're solving a math problem, preparing for competitive exams, or calculating patterns in real life, this formula makes the process quick and accurate. It is one of the most important tools in arithmetic sequences, commonly tested in CUET, JEE, SSC, Banking, and school-level exams.
nth Term of an Arithmetic Progression (AP) Overview
| Formula | Variables | When It Is Used |
|---|---|---|
| an = a + (n – 1)d | a = first term | To find any specific term in AP |
| d = common difference | Used in sequences & aptitude exams | |
| n = position of the term | Helpful for quick calculations |
What is the nth Term in Maths?
The nth term of an Arithmetic Progression (AP) tells us the value of a term at the nth position in the sequence. For example, in the series 2, 5, 8, 11…, the 4th term (n = 4) is 11.
In an AP, each number changes by a fixed amount known as the common difference (d). Instead of writing the full sequence to find a specific term, we use the formula:
an = a + (n – 1)d
Steps to Find the nth Term:
- Identify the first term (a).
- Calculate the common difference (d) by subtracting any two consecutive terms.
- Put the values of a, d, and n into the formula.
- Simplify to get the nth term.
The formula is helpful in real-life applications like calculating loan installments, savings patterns, scheduling, or numbering repeated patterns. It also appears frequently in competitive exams like CUET, SSC, Railway, JEE, and Banking tests.
Examples to Calculate nth Term of an AP
Example 1: Find the 15th term of the AP 3, 7, 11, …
Step 1: a = 3, d = 4, n = 15
Step 2: an = a + (n – 1)d
= 3 + (15 – 1) × 4
= 3 + 56
Result: 59
So, the 15th term of the AP is 59.
Example 2: Find the 25th term of the AP 10, 20, 30, …
Step 1: a = 10, d = 10, n = 25
Step 2: an = 10 + (25 – 1) × 10
= 10 + 240
Result: 250
So, the 25th term of the AP is 250.
FAQs about nth Term Formula
Q1. What is the nth term of the AP 5, 9, 13…?
Here a = 5 and d = 4.
So, an = 5 + (n – 1) × 4 = 4n + 1.
Q2. Can the nth term be negative?
Yes, if the AP decreases (negative common difference), the nth term can be negative.
Q3. Is nth term only used for AP?
No, nth term concepts exist in GP (Geometric Progression) and other sequences too.
Q4. What is the nth term of the first n natural numbers?
It is n, because the sequence is 1, 2, 3, …
Q5. Who introduced the concept of the nth term?
It comes from early number theory explored by ancient Greek mathematicians and later formalized in algebra.