Class 10 · Maths · Algebra

Arithmetic Progressions

4 min readUpdated November 2026

In one line

An arithmetic progression is a list of numbers where you add the same amount each time to get the next number.

Key points

Example: 3, 7, 11, 15, … Each term is 4 more than the one before.
a = the first term. d = the common difference = any term minus the term before it.
d can be positive (2, 5, 8 …), negative (20, 17, 14 …) or zero (6, 6, 6 …).
To check a list is an AP, make sure the difference is the same every time.

The nth term

aₙ = a + (n − 1)d

Example: find the 20th term of 3, 7, 11, …a = 3, d = 4. a₂₀ = 3 + 19 × 4 = 3 + 76 = 79.

Which term of 5, 8, 11, … is 101?101 = 5 + (n − 1) × 3 → 96 = 3(n − 1) → n − 1 = 32 → n = 33.

Sum of the first n terms

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

Here l is the last term. Use the second form when you already know it.

Sum of the first 100 natural numbersa = 1, l = 100, n = 100. S = 100/2 × (1 + 100) = 50 × 101 = 5050.

Savings problemRiya saves ₹50 in week 1 and ₹10 more each week. How much after 12 weeks?
a = 50, d = 10, n = 12. S = 12/2 × [100 + 11 × 10] = 6 × 210 = ₹1,260.

Useful fact: aₙ = Sₙ − Sₙ₋₁. If you know the sums, subtract to get a term.

Common mistakes

Using n instead of (n − 1). The 1st term is a, so the 20th term adds d only 19 times.
Getting d backwards. d = second term − first term, not the other way round. For 20, 17, 14 … d = −3.
Mixing up aₙ and Sₙ. aₙ is one term; Sₙ is the total of n terms.

Quick check

Revise in 30 seconds

Common difference	d = a₂ − a₁
nth term	aₙ = a + (n − 1)d
Sum	Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)
Term from sums	aₙ = Sₙ − Sₙ₋₁
1 + 2 + … + n	n(n + 1)/2