Arithmetic Progression (AP) Explained with Examples
An arithmetic progression, usually shortened to AP, is one of the most scoring chapters in CBSE and ICSE Class 10 maths, and it appears often in aptitude and competitive exams too. Once you understand two simple formulas, most questions become quick. This guide explains the common difference, the nth term formula and the sum formula with clear worked examples.
What is an arithmetic progression?
An arithmetic progression is a list of numbers in which the difference between any term and the term before it is always the same. That fixed gap is called the common difference, written as d. For example, 3, 7, 11, 15, 19 is an AP because each term is 4 more than the previous one, so d = 4.
- First term (a): the starting number of the sequence.
- Common difference (d): any term minus the term just before it, so d = second term − first term.
- General AP: a, a + d, a + 2d, a + 3d, and so on.
To check whether a list is an AP, subtract each term from the next. If you always get the same number, it is an AP. In 2, 5, 9, 14 the gaps are 3, 4, 5, so it is not an AP.
The nth term formula: a + (n − 1)d
The nth term of an AP is given by aₙ = a + (n − 1)d. This lets you find any term without writing out the whole list.
Example: Find the 10th term of 3, 7, 11, 15, … Here a = 3 and d = 4. So a₁₀ = 3 + (10 − 1) × 4 = 3 + 36 = 39. The 10th term is 39.
Example: Which term of the AP 5, 11, 17, … is 95? Here a = 5, d = 6. Set 95 = 5 + (n − 1) × 6, so 90 = 6(n − 1), giving n − 1 = 15 and n = 16. So 95 is the 16th term.
The sum formula for the first n terms
The sum of the first n terms is Sₙ = n/2 × [2a + (n − 1)d]. If you already know the last term l, you can use the shorter form Sₙ = n/2 × (a + l).
Example: Find the sum of the first 10 terms of 3, 7, 11, 15, … With a = 3 and d = 4, S₁₀ = 10/2 × [2 × 3 + (10 − 1) × 4] = 5 × [6 + 36] = 5 × 42 = 210.
Example: Add all whole numbers from 1 to 100. This is an AP with a = 1, l = 100, n = 100. Using the short form, S = 100/2 × (1 + 100) = 50 × 101 = 5050. This is the famous shortcut attributed to the mathematician Gauss.
Quick tips to solve AP questions faster
Always write down a, d and n first, then decide whether the question wants a term or a sum. If three numbers are in AP, the middle one equals the average of the other two, which is handy for finding unknowns. For more shortcuts like this, see our fun maths tricks for students, and if remembering formulas is your struggle, try our guide on how to memorise maths formulas. Students preparing the full syllabus can explore our Class 10 Maths resources for more practice.
Frequently asked questions
How do I find the common difference of an AP?
Subtract any term from the term that comes right after it. For the AP 8, 13, 18, the common difference is 13 − 8 = 5. You can check with another pair: 18 − 13 also equals 5, confirming it is a valid AP.
What is the difference between the nth term and the sum formula?
The nth term formula aₙ = a + (n − 1)d gives you the value of a single term at position n. The sum formula Sₙ = n/2 × [2a + (n − 1)d] adds up all terms from the first up to the nth term.
Can the common difference be negative or zero?
Yes. If d is negative the terms decrease, as in 20, 17, 14, 11 where d = −3. If d is zero, every term is the same, like 5, 5, 5, 5, which is still technically an AP.