Skip to main content
Class 10 · Chapter 5 · Algebra unit (20 of 80 marks)

Arithmetic Progressions Class 10: notes and important questions

Revision notes, 39 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.

  • 39 questions
  • 12 multiple choice, 3 assertion–reason, 8 very short answer, 8 short answer, 5 long answer, 3 case study
  • About 12 hours to master

Algebra unit: 20 of 80 theory marks (Polynomials, Pair of Linear Equations, Quadratic Equations, Arithmetic Progressions).

Revision notes

Arithmetic Progressions — revision notes

1. Definition

An AP is a list of numbers in which each term (after the first) is obtained by adding a fixed number \(d\), the common difference, to the previous term: \(a, a + d, a + 2d, \ldots\). \(d\) may be positive, negative or zero.

Three numbers \(p, q, r\) are in AP if and only if \(2q = p + r\).

2. General (\(n\)th) term

\[a_n = a + (n - 1)d\]

  • \(n\)th term from the end of an AP with last term \(l\): \(l - (n - 1)d\).
  • To test whether \(x\) is a term, solve \(a + (n-1)d = x\) and check that \(n\) is a natural number.

3. Sum of first \(n\) terms

\[S_n = \frac n2[2a + (n - 1)d] = \frac n2(a + l)\]

  • \(a_n = S_n - S_{n-1}\) (for \(n \geq 2\)); \(a_1 = S_1\).
  • Sum of first \(n\) natural numbers \(= \dfrac{n(n+1)}{2}\); first \(n\) odd numbers \(= n^2\).

4. Choosing terms

Three terms: \(a - d, a, a + d\) (sum \(= 3a\)). Useful when the sum is given.

Worked example 1

Which term of \(5, 9, 13, \ldots\) is \(101\)? \(5 + 4(n - 1) = 101 \Rightarrow n = 25\).

Worked example 2

Sum of all multiples of 4 between 10 and 250: \(12, 16, \ldots, 248\); \(n = \dfrac{248 - 12}{4} + 1 = 60\); \(S = 30(12 + 248) = 7800\).

Worked example 3

If \(S_n = 3n^2 - n\): \(a_1 = 2\), \(S_2 = 10 \Rightarrow a_2 = 8\), so \(d = 6\) and \(a_n = 6n - 4\).

Common errors

  • Using \(a + nd\) instead of \(a + (n-1)d\).
  • Finding the number of terms as \(\dfrac{l - a}{d}\) and forgetting the “\(+1\)”.
  • Confusing \(a_n\) (one term) with \(S_n\) (a sum).
  • Rejecting a second valid value of \(n\) when both give the same sum (negative terms can cancel).

Board-exam tips

  • Write \(a\), \(d\) and \(n\) clearly before substituting.
  • In word problems, first check that the situation really forms an AP (constant increase/decrease).
  • Large money answers: use the Indian system (₹4,56,000) and units.

Topics in this chapter: Arithmetic progressions and common difference · nth term of an AP · Sum of first n terms of an AP · Situational problems.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can spot an AP, find its common difference and any term, and count how many terms a list has.

Read first: 1. Definition; 2. General (nth) term; 3. Sum of first n terms 11 practice questions · checkpoint: 3 questions, 5 marks, pass 80%
Practise step 1
Step 2

Board standard

You can use Sₙ confidently, find a term from the end and solve the standard board questions on sums of multiples.

Read first: 2. General (nth) term; 3. Sum of first n terms; 4. Choosing terms; Worked examples 1-2 16 practice questions · checkpoint: 4 questions, 8 marks, pass 80%
Practise step 2
Step 3

Full marks on long answers

You can write complete 5-mark answers on AP word problems (savings, instalments, rows of seats) with a, d and n stated.

Read first: 3. Sum of first n terms; Board-exam tips 6 practice questions · checkpoint: 3 questions, 14 marks, pass 80%
Practise step 3
Step 4

95+ stretch (HOTS)

You can handle Sₙ given as a formula, 'three numbers in AP' and sums of the next n terms, the 95+ questions.

Read first: 4. Choosing terms; Worked example 3; Common errors 6 practice questions · checkpoint: 3 questions, 13 marks, pass 80%
Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 6 of the 39 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceArithmetic progressions and common difference

The common difference of the AP \(\dfrac73, 2, \dfrac53, \ldots\) is

  1. (a)\(\dfrac13\)
  2. (b)\(-1\)
  3. (c)\(-\dfrac13\)
  4. (d)\(\dfrac23\)
Q2·1 mark·Multiple choiceArithmetic progressions and common difference

If \(2k + 1,\ 13,\ 5k - 3\) are three consecutive terms of an AP, then \(k\) equals

  1. (a)\(3\)
  2. (b)\(5\)
  3. (c)\(4\)
  4. (d)\(\dfrac{27}{7}\)
Q3·1 mark·Multiple choicenth term of an AP

The \(15\)th term of the AP \(-4, -1, 2, \ldots\) is

  1. (a)\(-46\)
  2. (b)\(41\)
  3. (c)\(35\)
  4. (d)\(38\)

Where marks are lost in Arithmetic Progressions

  • Using a + nd for the nth term. Fix: aₙ = a + (n − 1)d; write the formula first so the M1 is safe.
  • Counting terms as (l − a)/d without the '+1'. Fix: n = (l − a)/d + 1, then check with a small example.
  • Confusing aₙ (one term) with Sₙ (a sum), especially when Sₙ is given as a formula. Fix: aₙ = Sₙ − Sₙ₋₁, and a₁ = S₁.
  • Wrongly rejecting a valid second value of n. Fix: when both roots are positive integers, check both; negative terms can make two sums equal.
  • Word problems where the story is not checked as an AP, or the answer lacks units. Fix: write 'increases by ₹50 each month, so AP with d = 50' and finish with ₹ in the Indian system.
  • Three-numbers-in-AP problems started as a, a + d, a + 2d. Fix: take a − d, a, a + d so the sum gives a in one step.

Arithmetic Progressions in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).