Class 10 · Chapter 5 · Algebra unit (20 of 80 marks)

Arithmetic Progressions Class 10: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 44 board-style questions.

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44 questions
13 multiple choice, 3 assertion–reason, 10 very short answer, 10 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).

NCERT textbook: Chapter 5 on ncert.nic.in (PDF)

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: $a, a + d, a + 2 d, \dots$ . $d$ may be positive, negative or zero.

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

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

$S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)$

$a_{n} = S_{n} - S_{n - 1}$ (for $n \geq 2$ ); $a_{1} = S_{1}$ .
Sum of first $n$ natural numbers $= \frac{n (n + 1)}{2}$ ; first $n$ odd numbers $= n^{2}$ .

4. Choosing terms

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

Worked example 1

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

Worked example 2

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

Worked example 3

If $S_{n} = 3 n^{2} - n$ : $a_{1} = 2$ , $S_{2} = 10 \Rightarrow a_{2} = 8$ , so $d = 6$ and $a_{n} = 6 n - 4$ .

Common errors

Using $a + n d$ instead of $a + (n - 1) d$ .
Finding the number of terms as $\frac{l - a}{d}$ 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.

Formulas for this chapter: Chapter 5 on the Class 10 formula sheet (free PDF and A3 poster).

Arithmetic Progressions Class 10: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 169 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choicenth term of an AP

The $15$ th term of the AP $7, 11, 15, \dots$ is

Show worked answer

$a = 7$ , $d = 4$ : $a_{15} = 7 + 14 \times 4 = 63$ A1
Correct option: (c) $63$

Q2·1 mark·Multiple choiceSum of first n terms of an AP

The sum $1 + 3 + 5 + \dots + 49$ equals

Q3·1 mark·Assertion–reasonArithmetic progressions and common difference

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): $\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \dots$ is an arithmetic progression.

Reason (R): Every term of an arithmetic progression must be a rational number.

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answernth term of an AP

Which term of the AP $5, 11, 17, \dots$ is $185$ ?

Show worked answer

Answer: the

31

st term

$a = 5$ , $d = 6$ : $5 + (n - 1) 6 = 185$ M1
$(n - 1) 6 = 180 \Rightarrow n = 31$ : the $31$ st term A1

Q5·2 marks·Short answerSum of first n terms of an AP

The $n$ th term of an AP is $a_{n} = 3 n + 2$ . Find the sum of its first $25$ terms.

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answernth term of an AP

In an AP, $a_{3} = 11$ and $a_{10} = 39$ . Find $a$ , $d$ and $a_{18}$ .

Show worked answer

Answer:

a = 3

d = 4

a_{18} = 71

$a + 2 d = 11$ and $a + 9 d = 39$ ; subtracting, $7 d = 28 \Rightarrow d = 4$ M1
$a = 11 - 8 = 3$ A1
$a_{18} = 3 + 17 \times 4 = 71$ A1

Q7·3 marks·Short answerSum of first n terms of an AP

How many three-digit numbers are divisible by $6$ ? Find their sum.

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerSum of first n terms of an AP

The sum of the first $n$ terms of an AP is $S_{n} = 3 n^{2} + 5 n$ . Find (a) the first term, (b) the common difference, (c) the $16$ th term, and (d) which term of the AP is $152$ .

Show worked answer

Answer:

8

;

6

;

98

; the

25

th term

(a) $a_{1} = S_{1} = 3 + 5 = 8$ A1
(b) $S_{2} = 12 + 10 = 22$ , so $a_{2} = 22 - 8 = 14$ and $d = 6$ A1
$a_{n} = S_{n} - S_{n - 1} = 6 n + 2$ M1
(c) $a_{16} = 6 (16) + 2 = 98$ A1
(d) $6 n + 2 = 152 \Rightarrow n = 25$ : the $25$ th term A1

Q9·5 marks·Long answerSituational problems

Riya saves ₹ $200$ in the first week of the year and then ₹ $50$ more each week than the week before. (a) In which week does she save ₹ $1000$ ? (b) How much does she save in the $20$ th week? (c) How much has she saved in all after $20$ weeks?

Section E style · Case study (4 marks)

Q10·4 marks·Case studySituational problems

Tree-planting drive. In a tree-planting drive, a volunteer group plants $20$ saplings on the first day and then $4$ more each day than on the day before, for $25$ days.

How many saplings are planted on day $10$ ? [1 mark]
On which day are $100$ saplings planted? [1 mark]
How many saplings are planted in the whole drive? [2 marks]

Show worked answer

$a_{10} = 20 + 9 \times 4 = 56$ A1
$20 + (n - 1) 4 = 100 \Rightarrow n = 21$ : day $21$ A1
Day $25$ : $a_{25} = 20 + 24 \times 4 = 116$ M1
$\begin{aligned} S_{25} & = \frac{25}{2} (20 + 116) \\ = 25 \times 68 \\ = 1700 \end{aligned}$ saplings A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 10 assertion–reason questions · the Class 10 syllabus checklist.

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. 8 of the 44 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 $\frac{7}{3}, 2, \frac{5}{3}, \dots$ is

(a) $\frac{1}{3}$
(b) $- 1$
(c) $- \frac{1}{3}$
(d) $\frac{2}{3}$

Q2·1 mark·Multiple choiceArithmetic progressions and common difference

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

(a) $3$
(b) $5$
(c) $4$
(d) $\frac{27}{7}$

Q3·1 mark·Multiple choicenth term of an AP

The $15$ th term of the AP $- 4, - 1, 2, \dots$ is

(a) $- 46$
(b) $41$
(c) $35$
(d) $38$

Worked solution·2 marks·Very short answerSituational problems

Tins are stacked in rows: $21$ tins in the bottom row, $2$ fewer in each row above, and $1$ tin in the top row. How many rows are there, and how many tins in all?

Full worked solution Free, no sign-in

$21, 19, \dots, 1$ : $1 = 21 + (n - 1) (- 2) \Rightarrow n = 11$ rows A1
$S_{11} = \frac{11}{2} (21 + 1) = 121$ tins A1

Every question here has a full worked solution. See them all →

Stretch yourself: original algebra problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I115 · Problem I122 · Problem I25 · Problem I107. All 29 →

Themed lesson: Ada Lovelace Day maths lesson · Halloween maths lesson · National Mathematics Day maths lesson (starter, activity and worked answers). All themed maths

Knowledge organiser: Arithmetic Progressions on one page (key facts, formulas and mistakes to avoid; printable PDF)

Formulas in this topic: nth term of an AP · Sum of n terms of an AP. All formulas A–Z

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.

Exam strategy: how answers are step-marked · competency-based questions

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).

Sample paper 1 (Board standard): questions 5, 20, 32, 36
Sample paper 2 (Board standard): questions 5, 6, 32
Sample paper 3 (Above board): questions 6, 7, 32, 38
Sample paper 4 (Above board): questions 6, 7, 27, 32
Sample paper 5 (95+ challenge): questions 6, 32

Want it against the clock? Take a timed 30-mark chapter test on Arithmetic Progressions, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Arithmetic Progressions: common questions

What is in CBSE Class 10 Maths Chapter 5, Arithmetic Progressions?

The topics are: Arithmetic progressions and common difference, nth term of an AP, Sum of first n terms of an AP, Situational problems. This page has revision notes for each, and 44 board-style practice questions with step-marked solutions.

How many marks is Arithmetic Progressions worth?

It is in the Algebra unit, which carries 20 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks 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.

Are the Arithmetic Progressions important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Arithmetic Progressions — revision notes

1. Definition

2. General (nth) term

3. Sum of first n terms

4. Choosing terms

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Arithmetic Progressions Class 10: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Arithmetic Progressions

Arithmetic Progressions in our sample papers

Arithmetic Progressions: common questions

Work through the Route to 95 with your progress saved

2. General ( $n$ th) term

3. Sum of first $n$ terms