Chapter 5: Arithmetic Progressions

Important questions — the ones the board asked most (20)

1. Q1Long answer5 marksAsked: 2026

   32. The third and ninth term of an A.P. are 4 and - 8 respectively. (i) Which term of the A.P. is zero ? (ii) Find the value of n if Sn = - 36.

   

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   Model answer · finding term and n, AP sum

   1. 1.Let the first term of the AP be a and the common difference be d. Using the formula for the nth term of an AP, an = a + (n - 1)d, the third term is a3 = a + 2d = 4 and the ninth term is a9 = a + 8d = -8.1 mark
   2. 2.Solving the two equations a + 2d = 4 and a + 8d = -8, subtracting the first from the second gives 6d = -12, so d = -2. Substituting d = -2 in a + 2d = 4 gives a - 4 = 4, so a = 8. Thus the first term is a = 8 and the common difference is d = -2.1 mark
   3. 3.To find which term of the AP is zero, we put an = 0 in a + (n - 1)d = 0, giving 8 + (n - 1)(-2) = 0. This gives 8 - 2n + 2 = 0, so 10 - 2n = 0, so n = 5. Therefore the fifth term of the AP is zero.1.5 marks
   4. 4.Using the formula for the sum of first n terms of an AP, Sn = n/2[2a + (n - 1)d], and substituting Sn = -36, a = 8 and d = -2, we get -36 = n/2[2(8) + (n - 1)(-2)] = n/2[16 - 2n + 2] = n/2[18 - 2n]. This simplifies to -72 = n(18 - 2n), i.e., -72 = 18n - 2n2, so 2n2 - 18n - 72 = 0, i.e., n2 - 9n - 36 = 0.1 mark
   5. 5.Solving the quadratic equation n2 - 9n - 36 = 0 by factorisation, we get (n - 12)(n + 3) = 0, so n = 12 or n = -3. Since n cannot be negative, the valid value is n = 12.0.5 marks

   • Write both equations a + 2d = 4 and a + 8d = -8 clearly before solving them.
   • State the values a = 8 and d = -2 explicitly after solving the equations.
   • Write the final quadratic equation n2 - 9n - 36 = 0 and reject n = -3 stating n cannot be negative.

   Marks split from a structure derived from past papers (checked twice); not CBSE’s official scheme

2. Q2Long answer5 marksAsked: 2026

   The third and ninth terms of an A.P. are 4 and -8 respectively. (i) Which term of this A.P. is zero? (ii) If S_n = -36, then find the value of n.

   

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   Model answer · AP nth term and sum

   1. 1.Given: the 3rd term of the AP is 4 and the 9th term is –8. Let a be the first term and d the common difference. We use an = a + (n – 1) d. a3 = a + (3 – 1) d a + 2d = 4 ...(1)0.5 marks
   2. 2.a9 = a + (9 – 1) d a + 8d = –8 ...(2)0.5 marks
   3. 3.Subtracting (1) from (2): 6d = –8 – 4 = –12 d = –2 Putting d = –2 in (1): a + 2(–2) = 4 a = 81 mark
   4. 4.(i) Let the nth term be zero, that is, an = 0. a + (n – 1) d = 0 8 + (n – 1)(–2) = 01 mark
   5. 5.2(n – 1) = 8 n – 1 = 4 n = 5 So, the 5th term of this AP is zero.1 mark
   6. 6.(ii) We use Sn = n/2 [2a + (n – 1) d] with a = 8, d = –2 and Sn = –36. –36 = n/2 [2 × 8 + (n – 1)(–2)] –36 = n/2 [16 + (n – 1)(–2)] –36 = n/2 (18 – 2n) = 9n – n²0.5 marks
   7. 7.n² – 9n – 36 = 0 (n – 12)(n + 3) = 0 n = 12 or n = –3 The number of terms cannot be negative, so we reject n = –3. Hence, n = 12.0.5 marks

   • Write both equations, "a + 2d = 4" and "a + 8d = –8", before you solve them for d = –2 and a = 8.
   • Finish part (i) with the sentence "the 5th term of this AP is zero".
   • In part (ii), factorise to (n – 12)(n + 3) = 0 and say why n = –3 is rejected before you write n = 12.

   Marks split as in the board’s published marking scheme

3. Q3Long answer5 marksAsked: 2024

   (A) The sum of the first and eighth terms of an Arithmetic Progression (A.P.) is 32 and their product is 60. Find the first term and common difference of this A.P. Also find the sum of the first 20 terms of this A.P. OR (B) The sum of the first 9 terms of an A.P. of 40 terms is 153 and the sum of its last 6 terms is 687. Find the first term and common difference of this A.P. Also find the sum of all the terms of this A.P.

   

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   Model answer · AP terms and sum

   1. 1.Answering option (A) of the OR choice (only one option is to be answered): the first term and the common difference of the AP are 2 and 4, or 30 and –4, and the sum of the first 20 terms is 800 or –160 respectively. The working follows. Given: the sum of the first term and the eighth term of the AP is 32, and their product is 60. Let a be the first term and d the common difference. The nth term is a + (n – 1)d, so a8 = a + 7d. a + a8 = 32 a + (a + 7d) = 32 2a + 7d = 32 ----- (i)1 mark
   2. 2.a × a8 = 60 a(a + 7d) = 60 ----- (ii)1 mark
   3. 3.From (i), a + 7d = 32 – a. Putting this into (ii): a(32 – a) = 60 a² – 32a + 60 = 0 (a – 2)(a – 30) = 0 So a = 2 or a = 30. When a = 2, (i) gives 4 + 7d = 32, so 7d = 28 and d = 4. When a = 30, (i) gives 60 + 7d = 32, so 7d = –28 and d = –4.2 marks
   4. 4.So the first term and the common difference of the AP are 2 and 4, or 30 and –4, respectively. The sum of the first n terms is Sn = n/2 [2a + (n – 1)d]. For a = 2 and d = 4: S20 = 20/2 [2 × 2 + (20 – 1) × 4] S20 = 10 (4 + 76) S20 = 8000.5 marks
   5. 5.For a = 30 and d = –4: S20 = 20/2 [2 × 30 + (20 – 1) × (–4)] S20 = 10 (60 – 76) S20 = 10 × (–16) S20 = –1600.5 marks

   • Write a8 = a + 7d, then number the two equations "2a + 7d = 32 ----- (i)" and "a(a + 7d) = 60 ----- (ii)".
   • Keep both pairs, "2 and 4, or 30 and –4", and work out S20 for each pair.
   • Write the formula Sn = n/2 [2a + (n – 1)d] before you put in the values for S20.

   Marks split as in the board’s published marking scheme

4. Q4Long answer5 marksAsked: 2024

   (A) The sum of first and eighth terms of an A.P. is 32 and their product is 60. Find the first term and common difference of the A.P. Hence, also find the sum of its first 20 terms. OR (B) In an A.P. of 40 terms, the sum of first 9 terms is 153 and the sum of last 6 terms is 687. Determine the first term and common difference of A.P. Also, find the sum of all the terms of the A.P.

   How the model answer begins

   1. Given: in an AP with first term a and common difference d, the sum of the first and eighth terms is 32, and their product is 60. The eighth term is a8 = a + 7d. a + a8 = 32 a + (a + 7d) = 32 2a + 7d = 32 ----- (i)

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5. Q5Long answer5 marksAsked: 2024

   35. A manufacturer of TV sets produced 720 TV sets in the fourth year and 880 TV sets in the eighth year. Assuming that the production increases uniformly by a fixed number every year, find the production in the tenth year and the total production in the first seven years.

   How the model answer begins

   1. Given: 720 TV sets were produced in the 4th year and 880 TV sets in the 8th year. The production increases uniformly by a fixed number every year, so the numbers of TV sets made in the 1st, 2nd, 3rd, ... years form an…

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6. Q6Long answer5 marksAsked: 2024

   The second term of an Arithmetic Progression is 29 and its fourth term is 51. If its last term is 425, find the total number of terms in this Arithmetic Progression and the sum of all these terms.

   How the model answer begins

   1. The nth term of an Arithmetic Progression is given by the formula aₙ = a + (n-1)d, where a is the first term and d is the common difference. Since the second term is 29, we get a₂ = a + (2-1)d, that is a + d = 29. Since…

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7. Q7Long answer5 marksAsked: 2024

   34. If the sum of the first 7 terms of an A.P. is 91 and that of the first 17 terms is 561, then find the sum of the first n terms and hence find the n^th term.

   How the model answer begins

   1. Given: the sum of the first 7 terms of the AP is 91 and the sum of the first 17 terms is 561. Let a be the first term and d the common difference. The sum of the first n terms of an AP is S = n/2 [2a + (n – 1) d]. S7 =…

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8. Q8Long answer5 marksAsked: 2023

   (a) The ratio of the 11th term to 17th term of an A.P. is 3 : 4. Find the ratio of 5th term to 21st term of the same A.P. Also, find the ratio of the sum of first 5 terms to that of first 21 terms. OR (b) 250 logs are stacked in the following manner : 22 logs in the bottom row, 21 in the next row, 20 in the row next to it and so on (as shown by an example). In how many rows, are the 250 logs placed and how many logs are there in the top row ?

9. Q9Long answer5 marksAsked: 2023

   (a) If the sum of the first 7 terms of an A.P. is -14 and that of 11 terms is -55, then find the sum of its first 'n' terms. OR (b) In an A.P., the sum of the first 'n' terms is 3n^2 + n. Find the first term and the common difference of the A.P. Hence, find its 15th term.

   How the model answer begins

   1. Given: in an AP, the sum of the first 7 terms is S7 = -14 and the sum of the first 11 terms is S11 = -55. To find: the sum of its first n terms, Sn. Let a be the first term and d be the common difference of the AP. The…

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10. Q10Long answer5 marksAsked: 2023

    (a) If the sum of the first 7 terms of an A.P. is -21 and that of the first 17 terms is -221, then find the sum of its first 'n' terms. OR (b) A man repays a loan of Rs 3,250 by paying Rs 20 in the first month and then increases the payment by Rs 15 every month. How long will it take to clear the loan?

    How the model answer begins

    1. (a) Given: the sum of the first 7 terms of the AP is S7 = –21, and the sum of the first 17 terms is S17 = –221. Let a be the first term and d the common difference of the AP. The sum of the first n terms of an AP is Sn…

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11. Q11Long answer5 marksAsked: 2023

    (a) The sum of the first 8 terms of an A.P. is 100 and the sum of its first 19 terms is 551. Find the sum of its first 'n' terms. OR (b) If the sum of the first p terms of an A.P. is the same as the sum of its first q terms, (p != q), then show that the sum of its first (p + q) terms is zero.

    How the model answer begins

    1. Solving part (a). Given: S8 = 100 and S19 = 551. Let a be the first term and d the common difference of the AP. The sum of the first n terms of an AP is given by S = n/2 [2a + (n – 1) d]. For n = 8: S8 = 8/2 (2a + 7d) =…

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12. Q12Long answer5 marksAsked: 2023

    (A) Find the sum of first 25 terms of the A.P. whose nth term is given by an = 5 + 6n. Also, find the ratio of 20th term to 45th term. OR (B) In an A.P., if Sn = 3n^2 + 5n and ak = 164, find the value of k.

    How the model answer begins

    1. (A) a₁ = 5 + 6 × 1 = 11 and d = 6 (each term is 6 more). OR (B) a₁ = S₁ = 3 + 5 = 8.

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13. Q13Long answer5 marksAsked: 2024

    In an A.P. if Sn = 4n^2 - n, then (i) find the first term and common difference. (ii) write the A.P. (iii) which term of the A.P. is 107 ?

    How the model answer begins

    1. (i) Given: the sum of the first n terms of the AP is Sn = 4n² – n. The sum of the first term alone is the first term itself, so putting n = 1: S1 = 4(1)² – 1 = 4 – 1 = 3 Therefore, the first term a = 3.

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14. Q14Long answer5 marksAsked: 2025

    (a) The sum of the third and seventh terms of an AP is 6 and their product is 8. Find the sum of the first 16 terms of this AP. OR (b) The minimum age of children to participate in a painting competition is 8 years. It was observed that the youngest child among the participants was 8 years old and the participants were seated in increasing order of their age with a common difference of 4 months. If the sum of the ages of all the participants is 168 years, then find the age of the eldest child participating in the painting competition.

    How the model answer begins

    1. Answer to (b): the age of the eldest child participating in the painting competition is 13 years, worked out below. Given: the youngest participant is 8 years old, each next participant is 4 months older, and the sum of…

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15. Q15Long answer5 marksAsked: 2025

    (a) An AP consists of 'n' terms whose n^th term is 4 and the common difference is 2. If the sum of 'n' terms of AP is -14, then find 'n'. Also, find the sum of the first 20 terms. OR (b) The sum of the first six terms of an arithmetic progression is 42. The ratio of the 10^th term to the 30^th term is 1 : 3. Calculate the first and the thirteenth terms of the AP.

16. Q16Long answer5 marksAsked: 2024

    How many terms of the arithmetic progression 27, 24, 21, ... should be taken so that their sum is 105? Which term of this progression is zero?

    How the model answer begins

    1. The sum is 105 when 5 or 14 terms are taken, and the 10th term of the AP is zero. Given: the AP 27, 24, 21, ... with Sn = 105. First term a = 27 Common difference d = 24 − 27 = −3

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17. Q17Long answer5 marksAsked: 2023

    How many terms of the arithmetic progression 45, 39, 33, ........ must be taken so that their sum is 180 ? Explain the double answer.

    How the model answer begins

    1. In the arithmetic progression 45, 39, 33, ..., the first term a = 45 and the common difference d = 39 - 45 = -6.

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18. Q18Long answer5 marksAsked: 2023

    Solve the equation for x : 1 + 4 + 7 + 10 + .... + x = 287

    How the model answer begins

    1. Given: 1 + 4 + 7 + 10 + .... + x = 287 The terms 1, 4, 7, 10, ... form an AP with first term a = 1 and common difference d = 4 – 1 = 3. Let the AP have n terms. Its last term is x, so using a_n = a + (n – 1)d: 1 + (n –…

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19. Q19Long answer5 marksAsked: 2023

    Prerna saves ₹ 32 during the first month, ₹ 36 in the second month and ₹ 40 in the third month. If she continues to save in this manner, in how many months will she save ₹ 2,000 ?

    How the model answer begins

    1. Prerna will save ₹ 2000 in 25 months. This is worked out below. Given: Prerna saves ₹ 32, ₹ 36 and ₹ 40 in the first three months, so she saves ₹ 4 more every month. Her monthly savings form an AP. According to the…

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20. Q20Long answer5 marksAsked: 2023

    (A) The ratio of the 11th term to the 18th term of an A.P. is 2 : 3. Find the ratio of the 5th term to the 21st term. Also, find the ratio of the sum of first 5 terms to the sum of first 21 terms. OR (B) If the sum of first 6 terms of an A.P. is 36 and that of the first 16 terms is 256, find the sum of first 10 terms.

    How the model answer begins

    1. (Attempting (B)) Given: for an AP, the sum of the first 6 terms is S6 = 36 and the sum of the first 16 terms is S16 = 256. Let a be the first term and d the common difference of the AP. The sum of the first n terms of…

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55 more questions of this chapter, each with its model answer, in the Ask Medha app.

Objective questions with the board’s key (10)

1. 1. CBSE 2026

   The common difference of the AP : sqrt(2), 2sqrt(2), 3sqrt(2), 4sqrt(2), ..... is :

   • (A) sqrt(2) ✓
   • (B) 1
   • (C) 2sqrt(2)
   • (D) -sqrt(2)

   Answer: (A)

2. 2. CBSE 2026

   The first term of an AP is p and the common difference is q, then its 10th term is :

   • (A) q - 9p
   • (B) p - 9q
   • (C) p + 9q ✓
   • (D) 2p + 9q

   Answer: (C)

3. 3. CBSE 2026

   Which of the following sequence is not an A.P. ?

   • (A) 2, 5/2, 3, 7/2, ...
   • (B) - 1.2, - 3.2, - 5.2, - 7.2, ...
   • (C) sqrt(2), sqrt(8), sqrt(18), ...
   • (D) 1^2, 3^2, 5^2, 7^2, ... ✓

   Answer: (D)

4. 4. CBSE 2026

   The value of x for which 2x, (x + 10) and (3x + 2) are the three consecutive terms of an A.P. is :

   • (A) 6 ✓
   • (B) - 6
   • (C) 18
   • (D) - 18

   Answer: (A)

5. 5. CBSE 2026

   Assertion (A) : The mean of first 'n' natural numbers is (n-1)/2. Reason (R): The sum of first 'n' natural numbers is n(n+1)/2.

   • (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
   • (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
   • (C) Assertion (A) is true, but Reason (R) is false.
   • (D) Assertion (A) is false, but Reason (R) is true. ✓

   Answer: (D)

6. 6. CBSE 2026

   The number of multiples of 4 lying between 12 and 250 is :

   • (A) 59 ✓
   • (B) 59.5
   • (C) 60
   • (D) 61

   Answer: (A)

7. 7. CBSE 2026

   The nth term of the A.P. -1/3, 2/3, 5/3, 8/3, ... is :

   • (A) 3n - 4
   • (B) n - 4/3 ✓
   • (C) (n - 2)/3
   • (D) (n - 4)/3

   Answer: (B)

8. 8. CBSE 2026

   The sum of first n terms of an A.P. is 50sqrt(2). If the first and the last terms are sqrt(2) and 19sqrt(2) respectively, then the value of n is :

   • (A) 10
   • (B) 5 ✓
   • (C) 15
   • (D) 20

   Answer: (B)

9. 9. CBSE 2026

   n^th term of the A.P. -3/2, 3/2, 9/2, ... is :

   • (A) (3n/2) - 3
   • (B) 3n - 9/2 ✓
   • (C) (3n - 9)/2
   • (D) 3n + 3/2

   Answer: (B)

10. 10. CBSE 2026

    nth term of an A.P. is 5n – 15. The common difference of the A.P. is :

    • (A) 5n
    • (B) 5 ✓
    • (C) – 5
    • (D) 10

    Answer: (B)

Questions parents and students ask

How many marks does "Arithmetic Progressions" carry in CBSE Class 10 Maths?

In the CBSE papers of 2023–2026, this chapter carried about 5 marks and about 2 questions a year; by marks it ranks 12 in Maths.

Which are the most important (VVI) questions of "Arithmetic Progressions"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 75 written questions of this chapter, with answers, are in the Ask Medha app.

Do the answers follow the board’s marking scheme?

Yes. CBSE publishes its marking schemes; every model answer here is written point by point to the marks the board’s own scheme gives, and checked by a second, independent model.