*nth Term of an ARITHMETIC PROGRESSION ( AP )*

nth term an of the AP with first term a and common difference d is given by a_{n} = a + (n – 1) d.

**nth Term from the end of an ARITHMETIC PROGRESSION ( AP**)

Let the last term of an AP be ‘l’ and the common difference of an AP is ‘d’ then the nth term from

the end of an AP is given by

ln = l – (n – 1) d.

**Q.1.** Find the 10th term of the AP : 2, 7, 12, . . .

**Q.2.** Which term of the AP : 21, 18, 15, . . . is – 81?

**Q.3.** Which term of the AP : 3, 8, 13, 18, . . . ,is 78?

**Q.4. **How many two-digit numbers are divisible by 3?

**Q.5.** How many three-digit numbers are divisible by 7?

**Q.6. **How many multiples of 4 lie between 10 and 250?

**Q.7.** Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.

**Q.8. **An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.

**Q.9.** If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?

**Q.10. **Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?

**Q.11.** Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.

**Q.12. **The sum of 4th term and 8th term of an AP is 24 and the sum of 6th and 10th terms is 44. Find

the AP.

**Q.13. **The sum of 5th term and 9th term of an AP is 72 and the sum of 7th and 12th terms is 97. Find

the AP.

**Q.14. **If the numbers n – 2, 4n – 1 and 5n + 2 are in AP, find the value of n.

**Q.15. **Find the value of the middle most term (s) of the AP : –11, –7, –3,…, 49.

**Q.16. **The sum of the first three terms of an AP is 33. If the product of the first and the third term

exceeds the second term by 29, find the AP.

**Q.17. **The sum of the 5th and the 7th terms of an AP is 52 and the 10th term is 46. Find the AP.

**Q.18. **Find the 20th term of the AP whose 7th term is 24 less than the 11th term, first term being 12.

**Q.19. **If the 9th term of an AP is zero, prove that its 29th term is twice its 19th term.

**Q.20.** The 26th, 11th and the last term of an AP are 0, 3 and –1/5 , respectively. Find the common

difference and the number of terms.

**Q.21. **Find whether 55 is a term of the AP: 7, 10, 13,— or not. If yes, find which term it is.

**Q.22.** Determine k so that k2+ 4k + 8, 2k2 + 3k + 6, 3k2 + 4k + 4 are three consecutive terms of an AP.

**Q.23. **Split 207 into three parts such that these are in AP and the product of the two smaller parts is

4623.

**Q.24. **The angles of a triangle are in AP. The greatest angle is twice the least. Find all the angles of the

triangle.

**Q.25.** If the nth terms of the two APs: 9, 7, 5, … and 24, 21, 18,… are the same, find the value of n.

Also find that term.

**Q.26. **If sum of the 3rd and the 8th terms of an AP is 7 and the sum of the 7th and the 14th terms is –3,

find the 10th term.

**Q.27. **Which term of the AP: 53, 48, 43,… is the first negative term?

**Q.28. **A sum of Rs 1000 is invested at 8% simple interest per year. Calculate the interest at the end of

each year. Do these interests form an AP? If so, find the interest at the end of 30 years making

use of this fact.

**Q.29. **In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so

on. There are 5 rose plants in the last row. How many rows are there in the flower bed?

**Q.30.** Find the 20th term from the last term of the AP : 3, 8, 13, . . ., 253.

**Q.31. **Find the 10th term from the last term of the AP : 4, 9 , 14, . . ., 254.

**Q.32. **Find the 6th term from the end of the AP 17, 14, 11, …… (–40).

**Q.33. **Find the 8th term from the end of the AP 7, 10, 13, …… 184.

**Q.34.** Find the 10th term from the last term of the AP : 8, 10, 12, . . ., 126.

**Q.35.** Find the 12th term from the end of the AP: –2, –4, –6,…, –100.

**Sum of First n Terms of an ARITHMETIC PROGRESSION ( AP )**

The sum of the first n terms of an AP is given by

where a = first term, d = common difference and n = number of terms.

where l = last term

**Q.36.** If the sum of the first 14 terms of an AP is 1050 and its first term is 10, find the 20th term.

**Q.37. **How many terms of the AP : 24, 21, 18, . . . must be taken so that their sum is 78?

**Q.38. **How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?

**Q.39. **Find the sum of first 24 terms of the list of numbers whose nth term is given by an = 3 + 2n

**Q.40. **Find the sum of the first 40 positive integers divisible by 6.

**Q.41. **Find the sum of the first 15 multiples of 8.

**Q.42. **Find the sum of the odd numbers between 0 and 50.

**Q.43. **Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.

**Q.44.** Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

**Q.45. **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.

**Q.46. **If an = 3 – 4n, show that a1,a2 ,a3 ,… form an AP. Also find S20 .

**Q.47. **In an AP, if Sn = n (4n + 1), find the AP.

**Q.48. **In an AP, if S_{n} = 3n^{2} + 5n and a_{k} = 164, find the value of k.

**Q.49. **If Sn denotes the sum of first n terms of an AP, prove that S_{12} = 3(S_{8} –S_{4})

**Q.50. **Find the sum of first 17 terms of an AP whose 4th and 9th terms are –15 and –30 respectively.

**Q.51.** If sum of first 6 terms of an AP is 36 and that of the first 16 terms is 256, find the sum of first 10

terms.

**Q.52. **Find the sum of all the 11 terms of an AP whose middle most term is 30.

**Q.53.** Find the sum of last ten terms of the AP: 8, 10, 12,—, 126.

**Q.54. **How many terms of the AP: –15, –13, –11,— are needed to make the sum –55? Explain the

reason for double answer.

**Q.55. **The first term of an AP is –5 and the last term is 45. If the sum of the terms of the AP is 120,

then find the number of terms and the common difference.

**Q.56. **Which term of the AP: –2, –7, –12,… will be –77? Find the sum of this AP upto the term –77.

**Q.57. **Find the sum of first seven numbers which are multiples of 2 as well as of 9.

**Q.58. **The sum of the first n terms of an AP whose first term is 8 and the common difference is 20 is

equal to the sum of first 2n terms of another AP whose first term is – 30 and the common

difference is 8. Find n.

**Q.59.** The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and

the last terms to the product of the two middle terms is 7 : 15. Find the numbers.

**Q.60. **The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is

**Q.61.** If the sum of the first ten terms of this AP is 235, find the sum of its first twenty terms.

**Q.62.** Find the sum of those integers between 1 and 500 which are multiples of 2 as well as of 5.

**Q.63. **Find the sum of those integers from 1 to 500 which are multiples of 2 as well as of 5 .

**Q.64.** Find the sum of those integers from 1 to 500 which are multiples of 2 or 5.

**Q.65. **The eighth term of an AP is half its second term and the eleventh term exceeds one third of its

fourth term by 1. Find the 15th term.

**Q.66. **An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the

last three is 429. Find the AP.

**Q.67. **Find the sum of the integers between 100 and 200 that are (i) divisible by 9 (ii) not divisible by 9

**Q.68. **The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the 5th term to the

21st term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms.

**Q.69. **Solve the equation : 1 + 4 + 7 + 10 +…+ x =287

**Q.70. **Solve the equation – 4 + (–1) + 2 +…+ x = 437

**Q.71. **Jaspal Singh repays his total loan of Rs. 118000 by paying every month starting with the first

installment of Rs 1000. If he increases the installment by Rs 100 every month, what amount will

be paid by him in the 30th installment? What amount of loan does he still have to pay after the

30th installment?

**Q.72. **The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x

such that the sum of the numbers of the houses preceding the house numbered x is equal to the

sum of the numbers of the houses following it. Find this value of x.

**Q.73. **A manufacturer of TV sets produced 600 sets in the third year and 700 sets in the seventh year.

Assuming that the production increases uniformly by a fixed number every year, find : (i) the

production in the 1st year (ii) the production in the 10th year (iii) the total production in first 7

years

**Q.74.** 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18

in the row next to it and so on. In how may rows are the 200 logs placed and how many logs are

in the top row?

**Q.75. **A contract on construction job specifies a penalty for delay of completion beyond a certain date

as follows: Rs 200 for the first day, Rs 250 for the second day, Rs 300 for the third day, etc., the

penalty for each succeeding day being Rs 50 more than for the preceding day. How much money

the contractor has to pay as penalty, if he has delayed the work by 30 days?

**Q.76.** A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall

academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each

of the prizes.

**Q.77.** In a school, students thought of planting 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 the same

as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of

Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How

many trees will be planted by the students?

**Q.78. **A spiral is made up of successive semicircles, with centres alternately at A and B, starting with

centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . . . What is the total length of such a spiral

made up of thirteen consecutive semicircles?

**Q.79.** 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. A

competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the

bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in

the same way until all the potatoes are in the bucket. What is the total distance the competitor has

to run?

**Q.80. **The students of a school decided to beautify the school on the Annual Day by fixing colourful

flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2

m. The flags are stored at the position of the middle most flag. Ruchi was given the

responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could

carry only one flag at a time. How much distance did she cover in completing this job and

returning back to collect her books? What is the maximum distance she travelled carrying a flag?

**Q.81. **Show that the sum of an AP whose first term is a, the second term b and the last term c, is equal

to