Arithmetic Progressions — nth Term and Sum Made Simple
Understand Arithmetic Progressions, the common difference, and the formulas for the nth term and the sum of n terms, explained with a staircase analogy, a fully worked example and a quick self-check quiz.
By the PadhoDost Team · 📖 7 min read · Updated 4 August 2026
Part of Class 10 (CBSE) prep🧠 Climbing an Even Staircase
Picture a staircase where every step rises by exactly the same height, say 15 cm. The 1st step is 15 cm above the floor, the 2nd is 30 cm, the 3rd is 45 cm, and so on. Each number is simply the one before it plus a fixed jump. That steady, equal jump is the whole idea behind an Arithmetic Progression — a list of numbers that grows or shrinks by the same amount every single time.
What Makes a Sequence an AP?
An Arithmetic Progression (AP) is a list of numbers in which the difference between any term and the term just before it is always the same. This fixed difference is called the common difference, written as d, and the first term is called a. To test whether a sequence is an AP, subtract each term from the next — if you always get the same number, it is an AP.
Adding Up the Terms
📝 Worked example: the 15th term and a sum
Consider the AP: 3, 7, 11, 15, ...
First term a = 3. Common difference d = 7 - 3 = 4.
Find the 15th term using a_n = a + (n - 1)d:
a_15 = 3 + (15 - 1) x 4 = 3 + 14 x 4 = 3 + 56 = 59
Now find the sum of the first 20 terms using S_n = (n/2)[2a + (n - 1)d]:
S_20 = (20/2) x [2 x 3 + (20 - 1) x 4] = 10 x [6 + 76] = 10 x 82 = 820
So the 15th term is 59 and the sum of the first 20 terms is 820.
| Quantity | Formula |
|---|---|
| Common difference | d = a_2 - a_1 |
| nth term | a_n = a + (n - 1)d |
| Sum of n terms | S_n = (n/2)[2a + (n - 1)d] |
| Sum using last term l | S_n = (n/2)(a + l) |
Remember these
- ✓An AP has a constant common difference d between consecutive terms.
- ✓nth term: a_n = a + (n - 1)d — use it to jump straight to any term.
- ✓Sum of n terms: S_n = (n/2)[2a + (n - 1)d].
- ✓If the last term l is known, S_n = (n/2)(a + l).
- ✓d can be positive (increasing AP), negative (decreasing AP), or zero (all terms equal).
⚡ Quick check
Which term of the AP 5, 8, 11, 14, ... is equal to 50?
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