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TutorVertexCBSE · CLASS 10 MATHEMATICS

CHAPTER 5 · OBJECTIVE PRACTICE

Arithmetic Progressions: MCQs with Answers

Arithmetic Progressions Case Study PYQs

44 selected questions from CBSE board examinations (2023–2026), including compartment, supplementary and second board papers, and official sample papers (sessions 2023–24 to 2026–27). Choose an answer, then open the solution when you need it.

Official answers: CBSE · Worked solutions: CBSE / Team TutorVertex, as labelled.

Topics and solution methods checked against NCERT Class 10 Mathematics, Chapter 5.

Practise 6 Arithmetic Progressions Assertion–Reason PYQs →

44 questions

Identifying an AP

QUESTION 1 · 1 MARK

Board exam · 2026 · Second board examination · Standard · 30/7/1 · Q7

Which sequence is an arithmetic progression?

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Official CBSE answer · Solution by Team TutorVertex

Formula used

d = a₂ − a₁ = a₃ − a₂ = … .

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

An AP must have equal differences between every pair of consecutive terms. Compare the differences in each listed sequence.

In an AP, the difference between consecutive terms is constant.

For 3, 3 + √2, 3 + 2√2, …, each difference is √2.

The other sequences do not have a constant difference.

Correct answer: B — 3, 3 + √2, 3 + 2√2, …

Common difference

QUESTION 2 · 1 MARK

Board exam · 2023 · Main · Basic · 430/5/1 · Q1

The eighth and fourteenth terms of an AP are 17 and 29. Find its common difference.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a₁₄ − a₈ = (14 − 8)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a₈ = 17 and a₁₄ = 29. The first term cancels when the two term formulas are subtracted.

a₁₄ − a₈ = (14 − 8)d

29 − 17 = 6d

d = 2.

Correct answer: B — 2

QUESTION 3 · 1 MARK

Board exam · 2023 · Main · Standard · 30/5/1 · Q3

An AP has nth term aₙ = 3n + 7. What is its common difference?

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Official CBSE answer · Solution by Team TutorVertex

Formula used

d = a₂ − a₁.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Use aₙ = 3n + 7 with n = 1 and n = 2 to find consecutive terms.

a₁ = 3(1) + 7 = 10; a₂ = 3(2) + 7 = 13.

d = a₂ − a₁ = 13 − 10 = 3.

Correct answer: B — 3

QUESTION 4 · 1 MARK

Board exam · 2024 · Main · Basic · 430/4/1 · Q10

In an AP, a₂₃ − a₁₉ = 32. Find the common difference.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a₂₃ − a₁₉ = (23 − 19)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a₂₃ − a₁₉ = 32. Subtracting the term formulas eliminates a.

a₂₃ − a₁₉ = 4d

4d = 32, so d = 8.

Correct answer: A — 8

QUESTION 5 · 1 MARK

Board exam · 2024 · Main · Standard · 30/3/1 · Q2

For x ≠ 0, find the common difference of the AP 12x, 1 − 4x2x, 1 − 8x2x, … .

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Official CBSE answer · Solution by Team TutorVertex

Formula used

d = a₂ − a₁.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a₁ = 12x and a₂ = 1 − 4x2x. The parameter x is non-zero, so the common denominator 2x is non-zero.

Subtract the first term from the second term.

d = 1 − 4x2x − 12x

d = −4x2x = −2.

Correct answer: B — −2

QUESTION 6 · 1 MARK

Board exam · 2025 · Main · Standard · 30/1/3 · Q17

The eleventh and thirteenth terms of an AP are 39 and 45. Find its common difference.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a₁₃ − a₁₁ = (13 − 11)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a₁₁ = 39 and a₁₃ = 45. Subtract the term formulas to eliminate a.

a₁₃ − a₁₁ = 2d

45 − 39 = 2d

d = 3.

Correct answer: D — 3

QUESTION 7 · 1 MARK

Board exam · 2023 · Compartment · Basic · 430/C/1 · Q1

An AP has common difference −6. Find a₂₀ − a₁₄.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a₂₀ − a₁₄ = (20 − 14)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here d = −6. The first term cancels when the two term formulas are subtracted.

a₂₀ − a₁₄ = (20 − 14)d

a₂₀ − a₁₄ = 6(−6) = −36.

Correct answer: C — −36

QUESTION 8 · 1 MARK

Board exam · 2026 · Second board examination · Basic · 430/8/1 · Q4

Find the common difference of the AP 2, −2, −6, −10, … .

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Official CBSE answer · Solution by Team TutorVertex

Formula used

d = a₂ − a₁.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a₁ = 2 and a₂ = −2. Subtract the first term from the second; keep the negative sign.

d = a₂ − a₁ = −2 − 2 = −4.

The next difference, −6 − (−2), is also −4.

Correct answer: C — −4

QUESTION 9 · 1 MARK

Board exam · 2026 · Second board examination · Standard · 30/7/2 · Q3

An AP has common difference 5. Find a₁₈ − a₁₃.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a₁₈ − a₁₃ = (18 − 13)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here d = 5. The first term cancels when the term formulas are subtracted.

a₁₈ − a₁₃ = (18 − 13)d

a₁₈ − a₁₃ = 5 × 5 = 25.

Correct answer: C — 25

QUESTION 10 · 1 MARK

Official sample paper · 2024–25 · Standard · 041 · Q4

The nth term of an AP is 7n − 4. Find the common difference.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

d = a₂ − a₁.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Use the given formula aₙ = 7n − 4 with n = 1 and n = 2 to obtain the first two terms.

a₁ = 7 − 4 = 3 and a₂ = 14 − 4 = 10.

d = a₂ − a₁ = 10 − 3 = 7.

Correct answer: A — 7

The nth term

QUESTION 11 · 1 MARK

Board exam · 2023 · Main · Basic · 430/1/1 · Q6

Find the seventh term of an AP with first term 28 and common difference −4.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = 28, d = −4 and n = 7.

a = 28, d = −4

a₇ = a + 6d = 28 + 6(−4) = 4.

Correct answer: B — 4

QUESTION 12 · 1 MARK

Board exam · 2023 · Main · Standard · 30/4/1 · Q3

Find the next term of the AP √6, √24, √54, … .

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

√24 = 2√6 and √54 = 3√6, so a = √6, d = √6 and n = 4.

The terms are √6, 2√6, 3√6, … .

Their common difference is √6.

The next term is 4√6 = √96.

Correct answer: B — √96

QUESTION 13 · 1 MARK

Board exam · 2024 · Main · Basic · 430/2/2 · Q18

Find the seventh term of an AP with first term −3 and common difference −2.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = −3, d = −2 and n = 7.

a₇ = a + 6d

a₇ = −3 + 6(−2) = −15.

Correct answer: D — −15

QUESTION 14 · 1 MARK

Board exam · 2024 · Main · Basic · 430/3/2 · Q6

Find the thirtieth term of the AP −3, −7, −11, … .

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = −3, d = −7 − (−3) = −4 and n = 30.

a = −3 and d = −4.

a₃₀ = −3 + 29(−4) = −119.

Correct answer: C — −119

QUESTION 15 · 1 MARK

Board exam · 2025 · Main · Basic · 430/4/1 · Q10

Find the twentieth term of the AP 10√2, 6√2, 2√2, … .

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = 10√2, d = 6√2 − 10√2 = −4√2 and n = 20.

a = 10√2 and d = −4√2.

a₂₀ = 10√2 + 19(−4√2)

a₂₀ = −66√2.

Correct answer: C — −66√2

QUESTION 16 · 1 MARK

Board exam · 2025 · Main · Basic · 430/5/1 · Q6

Find the tenth term of the AP −12, −19, −26, … .

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = −12, d = −19 − (−12) = −7 and n = 10.

a = −12 and d = −7.

a₁₀ = −12 + 9(−7) = −75.

Correct answer: A — −75

QUESTION 17 · 1 MARK

Board exam · 2026 · Main · Basic · 430/5/1 · Q10

Find the nth term of the AP −13, 43, 3, … .

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = −13 and d = 43 − (−13) = 53. Leave n as the term number to obtain the general term.

a = −13 and d = 43 − (−13) = 53.

aₙ = −13 + (n − 1)(53)

aₙ = 5n − 63.

Correct answer: B — 5n − 63

QUESTION 18 · 1 MARK

Board exam · 2025 · Supplementary · Basic · 430/S/2 · Q18

Find the twentieth term of the AP −4, −7, −10, −13, … .

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = −4, d = −7 − (−4) = −3 and n = 20.

a = −4 and d = −3.

a₂₀ = −4 + 19(−3) = −61.

Correct answer: C — −61

QUESTION 19 · 1 MARK

Official sample paper · 2026–27 · Basic · 241 · Q6

An AP has first term p and common difference q. Find its tenth term.

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Formula guide · Team TutorVertex

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = p, d = q and n = 10. The letters p and q stand for the first term and common difference respectively.

Official CBSE solution · typesetting normalised

a₁₀ = a + (10 − 1)d = p + 9q.

Correct answer: C — p + 9q

Finding the term number

QUESTION 20 · 1 MARK

Board exam · 2023 · Main · Basic · 430/2/1 · Q1

How many terms does the AP 14, 19, 24, 29, …, 119 contain?

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = 14, d = 19 − 14 = 5 and aₙ = 119. Find the term number n.

a = 14, d = 19 − 14 = 5

119 = 14 + (n − 1)5

105 = 5(n − 1), so n = 22.

Correct answer: C — 22

QUESTION 21 · 1 MARK

Board exam · 2024 · Main · Standard · 30/2/1 · Q16

Which term of the AP −29, −26, −23, …, 61 is 16?

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = −29, d = −26 − (−29) = 3 and aₙ = 16. Find n.

a = −29 and d = 3.

16 = −29 + (n − 1)3

45 = 3(n − 1), so n = 16.

Correct answer: B — 16th

QUESTION 22 · 1 MARK

Board exam · 2024 · Main · Standard · 30/5/1 · Q11

How many terms are in the AP 3, 6, 9, 12, …, 111?

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = 3, d = 3 and aₙ = 111. Find n.

a = 3 and d = 3.

111 = 3 + (n − 1)3 = 3n

n = 37.

Correct answer: C — 37

QUESTION 23 · 1 MARK

Board exam · 2024 · Compartment · Standard · 30/S/2 · Q11

The first and last terms of an AP are 7 and 73. Its sum is 480. Find the number of terms.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

Sₙ = n(a + l)2.

Sₙ is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term.

Here Sₙ = 480, a = 7 and l = 73. Find n.

Sₙ = n(a + l)2

480 = n(7 + 73)2 = 40n

n = 12.

Correct answer: B — 12

QUESTION 24 · 1 MARK

Board exam · 2025 · Supplementary · Standard · 30/S/2 · Q11

Which term of the AP 7, 10, 13, … is 52?

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = 7, d = 10 − 7 = 3 and aₙ = 52. Find n.

a = 7 and d = 3.

52 = 7 + (n − 1)3

45 = 3(n − 1), so n = 16.

Correct answer: C — 16th

QUESTION 25 · 1 MARK

Board exam · 2026 · Second board examination · Basic · 430/8/3 · Q8

How many terms are in the AP 5, 9, 13, …, 185?

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = 5, d = 9 − 5 = 4 and aₙ = 185. Find n.

a = 5 and d = 4.

185 = 5 + (n − 1)4

180 = 4(n − 1), so n = 46.

Correct answer: C — 46

QUESTION 26 · 1 MARK

Board exam · 2026 · Second board examination · Standard · 30/8/1 · Q6

Which term of the AP 21, 18, 15, … is zero?

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = 21, d = 18 − 21 = −3 and aₙ = 0. Find n.

a = 21 and d = −3.

0 = 21 + (n − 1)(−3)

3(n − 1) = 21, so n = 8.

Correct answer: C — 8th

QUESTION 27 · 1 MARK

Official sample paper · 2026–27 · Basic · 241 · Q7

Which term of the AP 21, 42, 63, 84, … is 210?

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Formula guide · Team TutorVertex

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = 21, d = 42 − 21 = 21 and aₙ = 210. Find n.

Official CBSE solution · typesetting normalised

210 = 21 + (n − 1)21 ⇒ n = 10.

Correct answer: B — 10th

Terms from the end & middle terms

QUESTION 28 · 1 MARK

Board exam · 2023 · Main · Standard · 30/5/1 · Q14

Find the eleventh term from the end of the AP 10, 7, 4, …, −62.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

Term at position k from the end = l − (k − 1)d.

l is the last term, d is the common difference in the original forward AP, and k is the positive integer position counted from the end.

Here l = −62, d = 7 − 10 = −3 and k = 11. Reading the sequence backwards changes the common difference to 3.

Starting from the end, the AP is −62, −59, −56, … .

Its first term is −62 and common difference is 3.

The eleventh term is −62 + 10(3) = −32.

Correct answer: C — −32

QUESTION 29 · 1 MARK

Board exam · 2024 · Main · Standard · 30/5/3 · Q9

Find the fourteenth term from the end of the AP −11, −8, −5, …, 49.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

Term at position k from the end = l − (k − 1)d.

l is the last term, d is the common difference of the original forward AP, and k is the position counted from the end.

Here l = 49, d = −8 − (−11) = 3 and k = 14.

Read the AP from the end: 49, 46, 43, … .

The fourteenth term is 49 + 13(−3) = 10.

Correct answer: B — 10

QUESTION 30 · 1 MARK

Official sample paper · 2024–25 · Basic · 241 · Q10

Find the fourth term from the end of the AP −8, −5, −2, …, 49.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

Term at position k from the end = l − (k − 1)d.

l is the last term, d is the common difference of the original forward AP, and k is the position counted from the end.

Here l = 49, d = −5 − (−8) = 3 and k = 4.

From the end, the AP is 49, 46, 43, 40, … .

Its fourth term is 49 + 3(−3) = 40.

Correct answer: B — 40

QUESTION 31 · 1 MARK

Official sample paper · 2026–27 · Standard · 041 · Q5

Find the middle term of the AP 10, 7, 4, …, −62.

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Formula guide · Team TutorVertex

aₙ = a + (n − 1)d.

For an odd number n of terms, the middle term is at position n + 12.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a = 10, d = 7 − 10 = −3 and the last term is −62. First find n, then the middle position.

Official CBSE solution · typesetting normalised

−62 = 10 + (n − 1)(−3) ⇒ n = 25.

Thus 25 + 12 = 13th term is the middle term.

a₁₃ = 10 + 12(−3) = −26.

Correct answer: A — −26

Sums of terms

QUESTION 32 · 1 MARK

Board exam · 2023 · Main · Standard · 30/6/2 · Q3

The first n terms of an AP have sum 3n² + n and its common difference is 6. Find its first term.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

a = S₁.

Sₙ is the sum of the first n terms, n is the number of terms, a is the first term, and d is the common difference.

Here Sₙ = 3n² + n. Put n = 1 to obtain the first term. The given common difference is d = 6.

The first term equals S₁.

S₁ = 3(1)² + 1 = 4.

Correct answer: D — 4

QUESTION 33 · 1 MARK

Board exam · 2024 · Main · Standard · 30/3/2 · Q17

The first n terms of an AP have sum 3n² + 4n and its common difference is 6. Find the first term.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

a = S₁.

Sₙ is the sum of the first n terms, n is the number of terms, a is the first term, and d is the common difference.

Here Sₙ = 3n² + 4n. Put n = 1 to find the first term. The given common difference is d = 6.

The first term equals S₁.

S₁ = 3(1)² + 4(1) = 7.

Correct answer: A — 7

QUESTION 34 · 1 MARK

Board exam · 2025 · Main · Basic · 430/6/1 · Q16

The sum of the first n terms of an AP is Sₙ = n(3n + 1)2. Find its first term.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

a = S₁.

Sₙ is the sum of the first n terms, n is the number of terms, a is the first term, and d is the common difference.

Here Sₙ = n(3n + 1)2. Substitute n = 1 to find the first term.

The first term equals S₁.

S₁ = 1(3 × 1 + 1)/2 = 2.

Correct answer: A — 2

QUESTION 35 · 1 MARK

Board exam · 2025 · Main · Standard · 30/1/1 · Q11

The sum of the first m terms of an AP is 2m² + 3m. Find its second term.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

a₂ = S₂ − S₁.

Sₘ is the sum of the first m terms; m is a positive integer count of terms. S₁ is the first term, S₂ is the sum of the first two terms, and a₂ is the second term.

The source uses m as the number of terms: Sₘ = 2m² + 3m. Find S₁ and S₂ by putting m = 1 and m = 2.

S₁ = 2(1)² + 3(1) = 5.

S₂ = 2(2)² + 3(2) = 14.

a₂ = S₂ − S₁ = 14 − 5 = 9.

Correct answer: B — 9

QUESTION 36 · 1 MARK

Board exam · 2026 · Main · Basic · 430/4/1 · Q7

An AP has sum 50√2, first term √2 and last term 19√2. How many terms does it contain?

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Official CBSE answer · Solution by Team TutorVertex

Formula used

Sₙ = n(a + l)2.

Sₙ is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term.

Here Sₙ = 50√2, a = √2 and l = 19√2. Find n.

Sₙ = n(a + l)2

50√2 = n(√2 + 19√2)2

50√2 = 10n√2, so n = 5.

Correct answer: B — 5

QUESTION 37 · 1 MARK

Board exam · 2026 · Main · Standard · 30/4/2 · Q17

In an AP, the first term is −3 and S₁₇ = 357. Find a₁₇.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

Sₙ = n(a + l)2.

Sₙ is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term. In this question l = a₁₇, the seventeenth term.

Here n = 17, a = −3, S₁₇ = 357 and l = a₁₇. Find the seventeenth term.

S₁₇ = 17(a + a₁₇)/2

357 = 17(−3 + a₁₇)/2

42 = −3 + a₁₇, so a₁₇ = 45.

Correct answer: C — 45

QUESTION 38 · 1 MARK

Board exam · 2025 · Supplementary · Standard · 30/S/1 · Q11

The first n terms of an AP have sum 2n² + 5n. Find its common difference.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

a = S₁, a₂ = S₂ − S₁ and d = a₂ − a.

Sₙ is the sum of the first n terms, n is the number of terms, a is the first term, and d is the common difference. a₂ denotes the second term.

Here Sₙ = 2n² + 5n. First find S₁ and S₂, then subtract consecutive terms.

S₁ = 2 + 5 = 7.

S₂ = 2(2)² + 5(2) = 18.

a₂ = S₂ − S₁ = 11.

d = a₂ − a₁ = 11 − 7 = 4.

Correct answer: B — 4

QUESTION 39 · 1 MARK

Official sample paper · 2025–26 · Basic · 241 · Q6

If 1 + 2 + 3 + … + 50 = 25k, find k.

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Formula guide · Team TutorVertex

Sₙ = n(n + 1)2 for 1 + 2 + … + n.

n is the number of consecutive natural numbers starting at 1, and Sₙ is their sum. This AP has first term a = 1 and common difference d = 1.

Here n = 50 and the question writes the sum as 25k. The unknown k is the number multiplying 25.

Official CBSE solution · typesetting normalised

50(51)/2 = 25k ⇒ k = 51.

Correct answer: B — 51

Finding unknown values

QUESTION 40 · 1 MARK

Board exam · 2024 · Main · Standard · 30/4/2 · Q18

The first three terms of an AP are 3p − 1, 3p + 5 and 5p + 1. Find p.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

d = a₂ − a₁ = a₃ − a₂.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Here a₁ = 3p − 1, a₂ = 3p + 5 and a₃ = 5p + 1; p is the unknown value. Equate the two consecutive differences.

The consecutive differences are equal.

(3p + 5) − (3p − 1) = (5p + 1) − (3p + 5)

6 = 2p − 4

2p = 10, so p = 5.

Correct answer: D — 5

QUESTION 41 · 1 MARK

Board exam · 2024 · Compartment · Standard · 30/S/3 · Q11

Two APs start with the same term. Their common differences are −3 and −5. Subtract the sixth term of the second AP from the sixth term of the first.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

First AP: aₙ = a + (n − 1)d₁.

Second AP: bₙ = a + (n − 1)d₂.

a is the common first term, n is the term number, d₁ and d₂ are the common differences of the first and second APs respectively, and aₙ and bₙ are their respective nth terms. In the general formula, d means the common difference of whichever AP is being considered.

Here n = 6, d₁ = −3 and d₂ = −5. Both APs have the same first term a. Find a₆ − b₆.

Let the common first term be a.

First AP: a₆ = a + 5(−3) = a − 15.

Second AP: b₆ = a + 5(−5) = a − 25.

a₆ − b₆ = (a − 15) − (a − 25) = 10.

Correct answer: D — 10

QUESTION 42 · 1 MARK

Board exam · 2024 · Compartment · visually impaired paper · Standard · 30(B)/S · Q1

In an AP, five times the fifth term equals nine times the ninth term. Find its fourteenth term.

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

Thus a₅ = a + 4d, a₉ = a + 8d and a₁₄ = a + 13d. The condition is 5a₅ = 9a₉.

5(a + 4d) = 9(a + 8d)

5a + 20d = 9a + 72d

a + 13d = 0

Therefore a₁₄ = 0.

Correct answer: C — 0

QUESTION 43 · 1 MARK

Official sample paper · 2023–24 · Standard · 041 · Q5

Two APs have the same common difference. Their first terms are −1 and −8. What is the first AP’s fourth term minus the second AP’s fourth term?

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Official CBSE answer · Solution by Team TutorVertex

Formula used

aₙ = a + (n − 1)d.

a is the first term, d is the common difference, n is the positive integer term number, and aₙ is the nth term.

The first AP starts at a = −1 and the second at a = −8. Both have the same d, and n = 4. Subtract the second fourth term from the first.

Let their common difference be d.

The fourth terms are −1 + 3d and −8 + 3d.

Their difference is (−1 + 3d) − (−8 + 3d) = 7.

Correct answer: C — 7

QUESTION 44 · 1 MARK

Official sample paper · 2025–26 · Basic · 241 · Q5

The expressions 2x, x + 10 and 3x + 2 are consecutive terms of an AP. Find x.

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Formula guide · Team TutorVertex

For three consecutive AP terms: 2a₂ = a₁ + a₃.

a₁, a₂ and a₃ denote the first, second and third of the three consecutive terms being considered. The middle term equals the average of its two neighbours.

Here a₁ = 2x, a₂ = x + 10 and a₃ = 3x + 2. The symbol x is the unknown value.

Official CBSE solution · typesetting normalised

2(x + 10) = (3x + 2) + 2x ⇒ x = 6.

Correct answer: A — 6

Frequently asked questions

How do I recognise an arithmetic progression?

The difference between consecutive terms must be constant. The common difference can be positive, zero or negative.

What is the difference between aₙ and Sₙ?

aₙ is the nth term. Sₙ is the sum of the first n terms. For n ≥ 2, aₙ = Sₙ − Sₙ₋₁; the first term is S₁.

Why are there no 2022 board MCQs in this collection?

Arithmetic Progressions belonged to Term 2 in the 2021–22 term-wise syllabus. The available 2022 main and compartment papers use written-answer questions. We do not turn these into MCQs or label sample questions as 2022 board MCQs.

Are case-study and Assertion–Reason questions included?

This page contains ordinary four-choice MCQs. The AP case-study questions identified in the reviewed sources require written answers and belong in a separate resource. Assertion–Reason questions are also kept separate.

Who wrote the solutions?

Where the official marking scheme includes mathematical working, we use CBSE’s working with normalised typesetting. Where it gives only the answer, Team TutorVertex supplies the explanation. Every solution identifies which applies. Added formula guides and symbol explanations are by Team TutorVertex and are labelled separately from official CBSE working.

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Arithmetic Progressions chapter · Quadratic Equations MCQs · Class 10 Maths resources

Key rules for this chapter
  • Symbols: a = first term, d = common difference, n = term number or number of terms, aₙ = nth term, Sₙ = sum of the first n terms, l = last term.
  • General form: a, a + d, a + 2d, … .
  • Common difference: d = a₂ − a₁ = a₃ − a₂ = … .
  • nth term: aₙ = a + (n − 1)d.
  • If the last term is l, the kth term from the end is l − (k − 1)d.
  • Sum: Sₙ = n[2a + (n − 1)d]/2 = n(a + l)2.
  • A term number n must be a positive integer.
  • Sum of the first n natural numbers: n(n + 1)2.
Question-paper sources & marking schemes

Use the year/session, examination, level, paper code and question number above each question to find its original. Links are collected here.

NCERT Class 10 Mathematics — Chapter 5: Arithmetic Progressions · Official 2021–22 term-wise syllabus

How this question bank is prepared

This selected collection contains 36 board-examination MCQs from 2023–2026 and 8 official sample-paper MCQs from sessions 2023–24 to 2026–27. Each included answer is checked against its corresponding official marking scheme and checked independently. Mathematical data and choice order are preserved; wording is lightly edited for clarity. Some repeated questions across sets are represented once. The collection is not an exhaustive list of all AP questions in every paper.

NCERT Chapter 5 is used to check chapter relevance, definitions, notation, formulas and solution methods. NCERT exercises are not presented as board questions. The 44 selected MCQs are text-only; diagrams accompanying written case-study or long-answer questions do not make those questions MCQs.

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