Chapter 5: Arithmetic Progressions

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

1. Q1Short answer2 marksAsked: 2022, 2025

   If 5th and 9th terms of an A.P. are 43 and 79 respectively, find the A.P.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 55 s

   Model answer · finding AP from two given terms

   1. 1.Using the formula for the nth term of an arithmetic progression, an = a + (n - 1)d, we get the equations for the given terms. For the 5th term: a + 4d = 43 For the 9th term: a + 8d = 79 Subtracting the first equation from the second: 4d = 36, so d = 91 mark
   2. 2.Putting d = 9 in the equation a + 4d = 43, we get a + 36 = 43, so a = 7. So the first term a = 7 and the common difference d = 9. The arithmetic progression is 7, 16, 25, 34, . . .1 mark

   • Write both equations a + 4d = 43 and a + 8d = 79 using an = a + (n - 1)d before solving.
   • Show the subtraction step clearly to get d = 9.
   • State a = 7, d = 9 and then write the full AP 7, 16, 25, 34, ... .

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

2. Q2Short answer2 marksAsked: 2017, 2022

   Which term of the A.P. 21, 18, 15, ... is -81?

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 55 s

   Model answer · nth term of an A.P.

   1. 1.The first term of the given AP 21, 18, 15, . . . is a = 21 and the common difference is d = 18 – 21 = – 3.1 mark
   2. 2.Using the formula a_n = a + (n – 1) d and putting a_n = – 81, a = 21, d = – 3, we get – 81 = 21 + (n – 1)(– 3), i.e., – 81 = 24 – 3n, i.e., – 105 = – 3n, so n = 35. Therefore, the 35th term of the given AP is – 81.1 mark

   • Write a = 21 and d = 18 - 21 = -3 clearly as the first step.
   • State the formula a_n = a + (n-1)d before substituting values.
   • Show the substitution step -81 = 21 + (n-1)(-3) leading to n = 35, and end with the conclusion that the 35th term is -81.

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

3. Q3Short answer2 marksAsked: 2025

   Find the sum of 3 + 11 + 19 + ... + 67.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 75 s

   Model answer · sum of AP given last term

   1. 1.The given list 3, 11, 19, ..., 67 is an arithmetic progression (AP) with first term a = 3, common difference d = 11 - 3 = 8, and last term l = 67.0.5 marks
   2. 2.Using the formula to find the nth term of an AP, l = a + (n - 1)d, we substitute the values: 67 = 3 + (n - 1) × 8. Solving this equation gives n = 9.0.5 marks
   3. 3.Since the first term and the last term of the AP are given, the sum of the first n terms is found using the formula S = n/2 (a + l).0.5 marks
   4. 4.Substituting the values a = 3, l = 67, and n = 9 in the formula S = n/2 (a + l), we get S = 9/2 (3 + 67) = 9/2 × 70 = 315. So, the required sum is 315.0.5 marks

   • Write a = 3, d = 8, l = 67 clearly stating this is an AP.
   • Show the substitution step 67 = 3 + (n-1) x 8 to get n = 9 before using the sum formula.
   • State the formula S = n/2 (a + l) before substituting values to get 315.

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

4. Q4Short answer2 marksAsked: 2026

   Total how many terms are there in the A.P. 6, 10, 14, 18, ..., 174 ?

   How the model answer begins

   1. In the given AP 6, 10, 14, 18, ..., 174, the first term a = 6 and the common difference d = 10 - 6 = 4.

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5. Q5Short answer2 marksAsked: 2026

   If the sum of three consecutive terms of an AP is 15 and product is 105 then find these terms.

   How the model answer begins

   1. Let the three consecutive terms of the arithmetic progression be (a-d), a, (a+d), where a is the middle term and d is the common difference.

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6. Q6Short answer2 marksAsked: 2026

   If third term of an A.P. is 4 and 9th term is - 8 then find the A.P.

   How the model answer begins

   1. The nth term of an arithmetic progression is given by a + (n - 1)d, where a is the first term and d is the common difference. Here the 3rd term is 4, so a + (3 - 1)d = 4, that is a + 2d = 4. The 9th term is -8, so a +…

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7. Q7Short answer2 marksAsked: 2026

   Find the sum of the first 12 terms of the A.P., 8, 3, - 2, - 7, ... .

   How the model answer begins

   1. Here, a = 8, d = 3 - 8 = -5.

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8. Q8Short answer2 marksAsked: 2026

   How many two-digit numbers are divisible by 5 ?

   How the model answer begins

   1. The list of two-digit numbers divisible by 5 is 10, 15, 20, ..., 95.

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9. Q9Short answer2 marksAsked: 2026

   If the sum of the first 14 terms of an A.P. is 1050 and its first term is 10 then find its common difference.

   How the model answer begins

   1. Here, S14 = 1050, n = 14, a = 10. We know that Sn = n/2[2a + (n-1)d]. Substituting the given values, 1050 = 14/2[2×10 + (14-1)d] = 14/2[20 + 13d].

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10. Q10Short answer2 marksAsked: 2022

    Find the 20th term of the A.P. 9, 13, 17, 21, ... .

    How the model answer begins

    1. The first term of the given AP 9, 13, 17, 21, . . . is a = 9 and the common difference is d = 13 - 9 = 4.

    The full answer from Medha Ma’am →
11. Q11Short answer2 marksAsked: 2022

    Find the sum of odd numbers lying between 0 and 50.

    How the model answer begins

    1. The odd numbers lying between 0 and 50 are 1, 3, 5, . . ., 49. This list forms an AP with first term a = 1 and common difference d = 2.

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12. Q12Short answer2 marksAsked: 2022

    How many three digit numbers are divisible by 7?

    How the model answer begins

    1. The three-digit numbers divisible by 7 form an AP. The first three-digit number divisible by 7 is 105 and the last three-digit number divisible by 7 is 994.

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13. Q13Short answer2 marksAsked: 2022

    Find the sum of first 22 terms of the A.P. 8, 3, -2, ...

    How the model answer begins

    1. For the AP 8, 3, -2, ..., the first term a = 8 and the common difference d = 3 - 8 = -5.

    The full answer from Medha Ma’am →
14. Q14Short answer2 marksAsked: 2022

    Which term of an A.P. 3, 10, 17, ... exceeds 84 from its 13th term ?

    How the model answer begins

    1. In the given AP 3, 10, 17, ..., the first term is a = 3 and the common difference is d = 10 - 3 = 7. The 13th term is given by a13 = a + 12d.

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15. Q15Short answer2 marksAsked: 2022

    What is the 10th term of an A.P. - 40, - 15, 10, 35, ... ?

    How the model answer begins

    1. For the given AP -40, -15, 10, 35, . . ., the first term a = -40 and the common difference d = (-15) - (-40) = 25.

    The full answer from Medha Ma’am →
16. Q16Short answer2 marksAsked: 2022

    Find the sum of first 40 positive integers which is divisible by 6.

    How the model answer begins

    1. The first 40 positive integers which are divisible by 6 form the list 6, 12, 18, . . ., 240. Here, 12 – 6 = 18 – 12 = 6 and so on, so this list is an AP.

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17. Q17Short answer2 marksAsked: 2023

    Find the common difference of an A.P. in which (a21 - a7) = 84.

    How the model answer begins

    1. The nth term of an AP is given by an = a + (n - 1) d, where a is the first term and d is the common difference. Using this formula, a21 = a + (21 - 1) d = a + 20d and a7 = a + (7 - 1) d = a + 6d. So a21 - a7 = (a + 20d)…

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18. Q18Short answer2 marksAsked: 2023

    Which term of the A.P. 7, 13, 19, ... is 205 ?

    How the model answer begins

    1. The first term a = 7 and the common difference d = 13 - 7 = 6.

    The full answer from Medha Ma’am →
19. Q19Short answer2 marksAsked: 2023

    Find the sum of 5 + 9 + 13 + 17 + ... + 81.

    How the model answer begins

    1. Here, the given list of numbers 5, 9, 13, 17, ..., 81 is an AP. The first term a = 5, the common difference d = 9 - 5 = 4, and the last term l = 81.

    The full answer from Medha Ma’am →
20. Q20Short answer2 marksAsked: 2023

    If the numbers (2n - 1), (3n + 2) and (6n - 1) are in A.P., find these numbers.

    How the model answer begins

    1. Since (2n – 1), (3n + 2) and (6n – 1) are in AP, the difference between consecutive terms is the same, so b is called the arithmetic mean of a and c, i.e., 2(3n + 2) = (2n – 1) + (6n – 1). On simplifying, 6n + 4 = 8n –…

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10 more questions of this chapter, each with its model answer, in the AskMedha app.

Objective questions with the board’s key (10)

1. 1. Bihar Board 2026

   If 3, x, y and 15 are in A.P. then x + y =

   • (A) 9
   • (B) 18 ✓
   • (C) 27
   • (D) 36

   Answer: (B)

2. 2. Bihar Board 2026

   Common difference of the sequence 5, 9, 13, 17, 21, ... is

   • (A) 5
   • (B) -4
   • (C) 4 ✓
   • (D) -5

   Answer: (C)

3. 3. Bihar Board 2026

   13th term of the sequence - 5, - 1, 3, 7, ... is

   • (A) 35
   • (B) 39
   • (C) 43 ✓
   • (D) 47

   Answer: (C)

4. 4. Bihar Board 2026

   Sum of the first n natural numbers is

   • (A) n(n+1)(2n+1)/6
   • (B) n(n+1)/2 ✓
   • (C) {n(n+1)/2}^2
   • (D) n(n-1)/2

   Answer: (B)

5. 5. Bihar Board 2026

   11 + 12 + 13 + ... + 45 =

   • (A) 970
   • (B) 980 ✓
   • (C) 990
   • (D) 1080

   Answer: (B)

6. 6. Bihar Board 2026

   The next term of the A.P. sqrt(7), sqrt(28), sqrt(63), ... is

   • (A) sqrt(70)
   • (B) sqrt(84)
   • (C) sqrt(98)
   • (D) sqrt(112) ✓

   Answer: (D)

7. 7. Bihar Board 2026

   How many terms of the A.P. 3, 7, 11, 15, .... will make the sum 406 ?

   • (A) 10
   • (B) 12
   • (C) 14 ✓
   • (D) 20

   Answer: (C)

8. 8. Bihar Board 2026

   The sum of first n terms of an A.P. is (5n - n^2). The nth term of the A.P. is

   • (A) 5 - 2n
   • (B) 6 - 2n ✓
   • (C) 2n - 5
   • (D) 2n - 6

   Answer: (B)

9. 9. Bihar Board 2026

   The common difference of the arithmetic progression 7, 7+sqrt(3), 7+2sqrt(3), 7+3sqrt(3) is

   • (A) 3
   • (B) sqrt(3) ✓
   • (C) 7
   • (D) 2

   Answer: (B)

10. 10. Bihar Board 2026

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

    • (A) -7, -5, -3, -1, ...
    • (B) 3, 5, 7, 9, ...
    • (C) 2, 4, 6, 8, ...
    • (D) 2^2, 4^2, 6^2, 8^2, ... ✓

    Answer: (D)

Questions parents and students ask

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

In the Bihar Board papers of 2021–2026, this chapter carried about 7 marks and about 11 questions a year; by marks it ranks 6 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 30 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

BSEB does not publish a marking scheme. Each answer is written point by point on a structure derived from past papers and checked by a second, independent model; it is never called the board’s official scheme.