Arithmetic Progression Cheat Sheet

Arithmetic Progression Cheat Sheet

Quick Revision

An arithmetic progression (AP) is a sequence in which each term differs from the one before it by a fixed number, the common difference d. With first term a, the terms are a, a + d, a + 2d, a + 3d, ….

1. The nth Term

aₙ = a + (n − 1)d

Here a is the first term, d the common difference and n the term number.

2. Sum of the First n Terms

Sₙ = n/2 × [2a + (n − 1)d]
Sₙ = n/2 × (a + l)        (l = last term)

Use the first form when you know a and d; use the second when you already know the last term l.

3. Arithmetic Mean

The arithmetic mean of two numbers a and b is their average:

AM = (a + b) / 2

If three numbers are in AP, the middle one is the AM of the other two.

4. Worked Example

For the AP 3, 7, 11, 15, … the first term is a = 3 and d = 4.

10th term  = 3 + (10 − 1) × 4 = 3 + 36 = 39
Sum of 10  = 10/2 × [2 × 3 + 9 × 4] = 5 × 42 = 210

5. Exam Quick Tips

• The common difference is later term − earlier term; it can be negative.
• Three numbers in AP are best written as a − d, a, a + d — their sum is 3a.
• To count the terms: n = (l − a)/d + 1.
• The sum formula needs only two of a, d, l and n.

Keep revising with the rest of the Maths section:

• All Maths topics
• Geometric Progression Cheat Sheet
• Average Cheat Sheet

How this was made: Drafted with AI assistance; spot-checked by the editor on 31 October 2026.