Arithmetic Progressions
An Arithmetic Progression (AP) is a list of numbers in which each term is obtained by adding a fixed number, the common difference, to the previous term. This chapter gives you two powerful formulas — one for the nth term and one for the sum of the first n terms — and is a reliable source of full marks in the CBSE board exam.
Learning objectives
- Recognise an AP and find its common difference.
- Use aₙ = a + (n − 1)d to find any term.
- Use the sum formula to add the first n terms.
- Find which term of an AP equals a given value.
- Apply AP formulas to real-life situations.
Key concepts
What makes a list an AP
A sequence is an AP if the difference between any term and the one before it is constant. This constant is the common difference d = aₙ − aₙ₋₁. The first term is denoted a.
The nth term
The general (nth) term of an AP with first term a and common difference d is aₙ = a + (n − 1)d. It lets you jump straight to any term without listing all the earlier ones.
Sum of the first n terms
The sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d]. If the last term l is known, it can also be written Sₙ = n/2 (a + l).
Important formulas
Common difference
d = aₙ − aₙ₋₁
nth term
aₙ = a + (n − 1)d
Sum of n terms
Sₙ = n/2 [2a + (n − 1)d]
Sum with last term
Sₙ = n/2 (a + l)
Key definitions
- Arithmetic Progression
- A sequence where each term differs from the previous by a fixed common difference.
- Common difference (d)
- The constant added to each term to get the next.
- First term (a)
- The starting term of the progression.
Solved examples
Q1. Find the 10th term of the AP 2, 7, 12, …
Solution: Here a = 2 and d = 5. a₁₀ = a + 9d = 2 + 9(5) = 47.
Q2. Find the sum of the first 20 terms of the AP 1, 3, 5, …
Solution: a = 1, d = 2, n = 20. S₂₀ = 20/2 [2(1) + 19(2)] = 10[2 + 38] = 10 × 40 = 400.
Q3. Which term of the AP 5, 11, 17, … is 95?
Solution: a = 5, d = 6. Set aₙ = 95: 5 + (n − 1)6 = 95 ⇒ (n − 1)6 = 90 ⇒ n − 1 = 15 ⇒ n = 16. So the 16th term is 95.
Common mistakes to avoid
- Using n instead of (n − 1) in the nth-term formula.
- Taking d as the second term rather than the difference of two consecutive terms.
- Forgetting the 1/2 in the sum formula.
- Mixing up aₙ (a single term) with Sₙ (the sum of terms).
Arithmetic Progressions — MCQ Quiz
12 questions with instant feedback. Use number keys 1–4 to answer.
The common difference of the AP 3, 7, 11, … is:
Practice questions
Short answer
Find the common difference of 100, 95, 90, …
d = 95 − 100 = −5.
Find the 5th term of the AP with a = 7 and d = 4.
a₅ = 7 + 4(4) = 23.
Is 0, 0, 0, … an AP?
Yes, with first term 0 and common difference 0.
Long answer
Find the sum of the first 25 terms of the AP whose nth term is 3n + 1.
First term a = 3(1) + 1 = 4, and the 25th term = 3(25) + 1 = 76. S₂₅ = 25/2 (a + l) = 25/2 (4 + 76) = 25/2 × 80 = 1000.
How many terms of the AP 9, 17, 25, … add up to 636?
a = 9, d = 8. Sₙ = n/2[18 + (n − 1)8] = 636 ⇒ n/2[8n + 10] = 636 ⇒ 4n² + 5n − 636 = 0. Solving, n = 12 (rejecting the negative root).
HOTS (Higher Order Thinking)
The 7th term of an AP is 34 and the 13th term is 64. Find the AP.
a + 6d = 34 and a + 12d = 64. Subtracting: 6d = 30 ⇒ d = 5, then a = 34 − 30 = 4. The AP is 4, 9, 14, 19, …
If the sum of the first n terms of an AP is 3n² + 5n, find its 10th term.
Sₙ = 3n² + 5n, so a₁₀ = S₁₀ − S₉ = (300 + 50) − (243 + 45) = 350 − 288 = 62.
Quick revision
Revision notes
- AP: each term = previous term + d.
- nth term: aₙ = a + (n − 1)d.
- Sum: Sₙ = n/2[2a + (n − 1)d] = n/2(a + l).
- d = difference of any two consecutive terms.
Key takeaways
- Always identify a and d first.
- Use Sₙ = n/2(a + l) when the last term is known — it's quicker.
- aₙ = Sₙ − Sₙ₋₁ links the term and sum formulas.
Frequently asked questions
Is Arithmetic Progressions a scoring chapter?
Yes. The two formulas cover most questions, and the chapter regularly appears in the CBSE board exam.
How do I find which term equals a given number?
Set aₙ = a + (n − 1)d equal to that number and solve for n.
What is the difference between aₙ and Sₙ?
aₙ is a single term (the nth one); Sₙ is the total of the first n terms.