Chapter 5: Arithmetic Progressions
CBSE Official PYQs (2015-2026)
Q1. If the common difference of an AP is 5, then what is \(a_{18} - a_{13}\)?
Q2. In an AP, if \(a = 15\), \(d = -3\), \(n = 8\), then the \(n^{\text{th}}\) term \(a_n\) is :
Q3. The \(11^{\text{th}}\) term of the AP: \(-3, -\frac{1}{2}, 2, \dots\) is :
Q4. Which term of the AP: \(21, 18, 15, \dots\) is \(-81\)?
Q5. The sum of the first 16 terms of the AP: \(10, 6, 2, \dots\) is :
Q6. If the \(p^{\text{th}}\) term of an AP is \(q\) and \(q^{\text{th}}\) term is \(p\), then its \(n^{\text{th}}\) term is :
Q7. If the sum of the first \(n\) terms of an AP is \(3n^2 + n\), then its common difference is :
Q8. The \(10^{\text{th}}\) term of the AP: \(5, 8, 11, 14, \dots\) is :
Q9. The common difference of the AP whose \(n^{\text{th}}\) term is \(a_n = 3n + 7\) is :
Q10. The next term of the AP: \(\sqrt{7}, \sqrt{28}, \sqrt{63}, \dots\) is :
Q11. The first four terms of an AP whose first term is \(-2\) and common difference is \(-2\) are :
Q12. If \(k, 2k-1\) and \(2k+1\) are three consecutive terms of an AP, then the value of \(k\) is :
Q13. In an AP, if \(d = -4\), \(n = 7\), and \(a_n = 4\), then the first term \(a\) is :
Q14. The list of numbers \(-10, -6, -2, 2, \dots\) is :
Q15. The sum of the first 20 odd natural numbers is :
Q16. Which term of the AP: \(3, 8, 13, 18, \dots\) is \(78\)?
Q17. The \(21^{\text{st}}\) term of the AP whose first two terms are \(-3\) and \(4\) is :
Q18. If the common difference of an AP is \(-4\), and the seventh term (\(a_7\)) is \(4\), then the first term (\(a\)) is :
Q19. If the \(n^{\text{th}}\) term of an AP is given by \(a_n = 5 - 3n\), then the sum of its first 10 terms is :
Q20. The sum of the first five multiples of 3 is :
Q21. If the \(17^{\text{th}}\) term of an AP exceeds its \(10^{\text{th}}\) term by 7, then the common difference is :
Q22. For what value of \(p\) are \(2p+1\), \(13\), \(5p-3\) three consecutive terms of an AP?
Q23. The \(9^{\text{th}}\) term of the AP: \(\frac{1}{m}, \frac{1+m}{m}, \frac{1+2m}{m}, \dots\) is :
Q24. The sum of first \(n\) terms of an AP is \(5n - n^2\). The \(n^{\text{th}}\) term of this AP is :
Q25. The first and last terms of an AP are 1 and 11. If the sum of its terms is 36, then the number of terms is :
Q26. If \(a, b, c\) are in AP, then \(b-a\) is equal to :
Q27. Two APs have the same common difference. The first term of one of these is \(-1\) and that of the other is \(-8\). Then the difference between their \(4^{\text{th}}\) terms is :
Q28. If the common difference of an AP is 3, then \(a_{20} - a_{15}\) is :
Q29. If 7 times the \(7^{\text{th}}\) term of an AP is equal to 11 times its \(11^{\text{th}}\) term, then its \(18^{\text{th}}\) term is :
Q30. The sum of the first 100 positive integers is :
Q31. Find the number of terms in the AP: \(7, 13, 19, \dots, 205\).
Q32. Find the \(20^{\text{th}}\) term from the last term of the AP: \(3, 8, 13, \dots, 253\).
Q33. How many two-digit numbers are divisible by 3?
Q34. If the sum of the first \(n\) terms of an AP is given by \(S_n = 2n^2 + 3n\), find its \(10^{\text{th}}\) term.
Q35. In an AP, if the \(3^{\text{rd}}\) term is 5 and the \(7^{\text{th}}\) term is 9, find the AP.
Q36. Find the sum of first 20 terms of an AP whose first term is 5 and common difference is 4.
Q37. Which term of the AP: \(121, 117, 113, \dots\) is its first negative term?
Q38. Find the middle term of the AP: \(6, 13, 20, \dots, 216\).
Q39. If \(S_n\) denotes the sum of first \(n\) terms of an AP, prove that \(S_{12} = 3(S_8 - S_4)\).
Q40. Find the sum of all eleven terms of an AP whose middle term is 30.
Q41. The sum of the \(4^{\text{th}}\) and \(8^{\text{th}}\) terms of an AP is 24 and the sum of the \(6^{\text{th}}\) and \(10^{\text{th}}\) terms is 44. Find the first three terms of the AP.
Q42. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first \(n\) terms.
Q43. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
Q44. If the \(m^{\text{th}}\) term of an AP is \(1/n\) and \(n^{\text{th}}\) term is \(1/m\), show that the sum of first \(mn\) terms is \((mn+1)/2\).
Q45. If the sum of first \(p\) terms of an AP is equal to the sum of its first \(q\) terms (where \(p \neq q\)), then find the sum of its first \((p+q)\) terms.
Q46. Find the sum of all three-digit natural numbers which are multiples of 7.
Q47. The sum of the first \(n\) terms of an AP whose first term is 8 and common difference is 20 is equal to the sum of first \(2n\) terms of another AP whose first term is -30 and common difference is 8. Find \(n\).
Q48. Solve the equation: \(1 + 4 + 7 + 10 + \dots + x = 287\).
Q49. Case Study: Monthly Savings Plan
A woman plans to save money for her daughter's higher education. She saves Rs. 2000 in the first month, Rs. 2100 in the second month, Rs. 2200 in the third month, and so on.
Based on the above information, answer the following questions:
- Formulate an AP representing her monthly savings and find her savings in the \(24^{\text{th}}\) month. (2 Marks)
- Find the total amount saved by her at the end of 2 years (24 months). (2 Marks)
1. AP: 2000, 2100, 2200, ...; Savings in 24th month = Rs. 4300
2. Total savings in 24 months = Rs. 75,600
Q50. Case Study: Auditorium Seating Arrangement
In an auditorium, seats are arranged in rows where each row has a fixed number of seats more than the previous row.
- Row 1 has 30 seats
- Row 2 has 34 seats
- Row 3 has 38 seats, and so on...
Based on the above information, answer the following questions:
- Formulate an AP representing the seating arrangement and find the common difference. (1 Mark)
- Find the number of seats in the \(15^{\text{th}}\) row. (2 Marks)
- If there are 1500 total seats in the auditorium, find the total number of rows required. (2 Marks)
1. AP: 30, 34, 38, ... ; Common difference d = 4
2. Number of seats in 15th row \(a_{15} = 86\) seats
3. Total 22 rows are required to accommodate 1500 seats.
Q51. Case Study: Potato Race
In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.
Based on the above information, answer the following questions:
- Find the distance run by a competitor to pick up the \(3^{\text{rd}}\) potato and drop it back in the bucket. (2 Marks)
- Find the total distance covered by the competitor to collect all the ten potatoes. (3 Marks)
1. Distance for 3rd potato = 22 m
2. Total distance covered to collect all ten potatoes = 370 m
Q52. Case Study: Production of TV Sets
A manufacturer of TV sets produced 600 sets in the \(3^{\text{rd}}\) year and 700 sets in the \(7^{\text{th}}\) year. Assuming that the production increases uniformly by a fixed number every year:
Based on the above information, answer the following questions:
- Find the production in the \(1^{\text{st}}\) year. (1 Mark)
- Find the production in the \(10^{\text{th}}\) year. (2 Marks)
1. Production in 1st year = 550 TV sets
2. Production in 10th year = 775 TV sets
3. Total production in 7 years = 4375 TV sets