Arithmetic Progression in Mathematics

Updated on July 14, 2025 | By Learnzy Academy

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference and is usually denoted by d.

General Form of an AP:

a, a + d, a + 2d, a + 3d, ..., a + (n - 1)d

Where:

  • a = the first term
  • d = the common difference
  • n = the number of terms
  • a + (n - 1)d = the nth term (also written as Tn)

 

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List of question on "Arithmetic Progression"

  1. If 1 + 4 + 7 + 10 + ⋯ + x = 287, find the value of x.
  2. A child puts one five–rupee coin of her saving in the piggy bank on the first day. She increases her saving by one five–rupee coin daily. If the piggy bank can hold 190 coins of five rupees in all, find the number of days she can continue to put the five–rupee coins into it and find the total money she saved. Write your views on the habit of saving.
  3. The ratio of the sums of the first m and first n terms of an arithmetic progression is m² : n². Show that the ratio of its mth and nth terms is (2m − 1) : (2n − 1).
  4. If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) ∶ (4n + 27), then find the ratio of their 9th terms.
  5. The minimum age of children to be eligible to participate in a painting competition is 8 years. It is observed that the age of youngest boy was 8 years and the ages of rest of participants are having a common difference of 4 months. If the sum of ages of all the participants is 168 years, find the age of eldest participant in the painting competition.
  6. Reshma wanted to save at least ₹ 6,500 for sending her daughter to school next year (after 12 months). She saved ₹ 450 in the first month and raised her saving by ₹ 20 every next month. How much will she be able to save in next 12 months? Will she be able to send her daughter to the school next year? What value is reflected in this question?
  7. A thief runs with a uniform speed of 100 m/minute. After one minute a policeman runs after the thief to catch him. He goes with a speed of 100 m/minute in the first minute and increases his speed by 10 m/minute every succeeding minute. After how many minutes the policeman will catch the thief.
  8. The house in a row are numbered consecutively from 1 to 49. Show that there exists a value of X such that sum of numbers of houses preceeding the house numbered X is equal to sum of the numbers of houses following X.
  9. Yasmeen saves Rs.32 during the first month, Rs.36 in the second month and Rs.40 in the third month. If she continues to save in this manner, in how many months she will save Rs.2000, which she has intended to give for the college fee of her maid’s daughter. What value is reflected here.
  10. Find the 60th term of the AP 8, 10, 12, ..., if it has a total of 60 terms and hence find the sum of its last 10 terms.
  11. Find the middle term of the sequence formed by all three–digit numbers which leave a remainder 3, when divided by 4. Also find the sum of all numbers on both sides of the middle term separately.
  12. Ramkali required ₹ 2500 after 12 weeks to send her daughter to school. She saved ₹ 100 in the first week and increased her weekly saving by ₹ 20 every week. Find whether she will be able to send her daughter to school after 12 weeks. What value is generated in the above situation?
  13. In a school, students decided to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be double of the class in which they are studying. If there are 1 to 12 classes in the school and each class has two sections, find how many trees were planted by the students. Which value is shown in this question?
  14. If the sum of first 7 terms of an AP is 49 and that of first 17 terms is 289, find the sum of its first n terms.
  15. Find the common difference of an A.P. whose first term is 5 and the sum of its first four terms is half the sum of the next four terms.
  16. The sum of 4th and 8th terms of an A.P. is 24 and the sum of its 6th and 10th terms is 44. Find the sum of first ten terms of the A.P.
  17. Sum of the first 14 terms of an AP is 1505 and its first term is 10. Find its 25th term.
  18. The first and the last terms of an A.P. are 8 and 350 respectively. If its common difference is 9, how many terms are there and what is their sum?
  19. If the common difference of an AP is 3, then what is a15 − a9 ?
  20. For what value of k will k + 9, 2k − 1 and 2k + 7 are the consecutive terms of an A.P.?
  21. Find the 9th term from the end (towards the first term) of the A.P. 5,9,13, ... ... ,185.
  22. For what value of k will the consecutive terms 2k + 1, 3k + 3 and 5k − 1 form an A.P.?
  23. Find the tenth term of the sequence √2, √8, √18, ...
  24. What is the common difference of an arithmetic progression (A.P.) in which a₍₂₁₎ − a₍₇₎ = 84?
  25. Find the 17th term from the end of the AP: 1, 6, 11, 16 ... ... 211, 216.
  26. Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?
  27. Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?
  28. The 17th term of an AP exceeds its 10th term by 7. Find the common difference
  29. If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?
  30. An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.
  31. Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73
  32. Check whether – 150 is a term of the AP : 11, 8, 5, 2 . . .
  33. Which term of the given AP : 3, 8, 13, 18, . . . ,is 78?
  34. Which term of the AP : 21, 18, 15, . . . is – 81? Also, is any term 0? Give reason for your answer.
  35. Find the value of m so that m + 2, 4m – 6 and 3m – 2 are three consecutive terms of an AP.
  36. How many multiples of 4 lie between 10 and 250? Also find their sum.
  37. In an A.P., if the 6th and 13th terms are 35 and 70 respectively, find the sum of its first 20 terms.
  38. The 4ᵗʰ term of an A.P. is zero. Prove that the 25ᵗʰ term of the A.P. is three times its 11ᵗʰ term.
  39. If seven times the 7th term of an A.P. is equal to eleven times the 11th term, then what will be its 18th term?
  40. Find the sum of all natural numbers that are less than 100 and divisible by 4.
  41. The sum of all two digit odd numbers is
  42. If p, q, r and s are in A.P. then r – q is
  43. If the sum of three numbers in an A.P. is 9 and their product is 24, then numbers are
  44. Which term of the A.P. 3, 8, 13, 18, … is 78?
  45. The 21st term of AP whose first two terms are -3 and 4 is:
  46. If 17th term of an A.P. exceeds its 10th term by 7. The common difference is:
  47. The number of multiples of 4 between 10 and 250 is:
  48. 20th term from the last term of the A.P. 3, 8, 13, …, 253 is:
  49. The sum of the first five multiples of 3 is:
  50. If the 2nd term of an AP is 13 and the 5th term is 25, then its 7th term is
  51. Which term of the AP: 21, 42, 63, 84,… is 210 ?
  52. A person saves ₹500 in the first month and increases savings by ₹100 every month. How much does he save in the 12th month?
  53. A ladder has rungs placed at equal distances. If the first rung is at 15 cm and the last at 120 cm, with 10 rungs, find the distance between two consecutive rungs.
  54. If a train travels 5 km in the first minute and increases speed by 2 km per minute, how much distance will it cover in 20 minutes?
  55. The sum of first 16 terms of the AP: 10, 6, 2,… is