# Arithmetic Progressions

**Exercise 1**

**1.** In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?

(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km.

(ii) The amount of air present in a cylinder when a vacuum pump removes 14th of the air remaining in the cylinder at a time.

(iii) The cost of digging a well after every meter of digging, when it costs Rs 150 for the first meter and rises by Rs 50 for each subsequent meter.

(iv) The amount of money in the account every year, when Rs 10,000 is deposited at compound Interest at 8% per annum.

**2.** Write first four terms of the AP, when the first term a and common difference d are given as follows:

(i) a = 10, d = 10 (ii) a = -2, d = 0

(iii) a = 4, d = -3 (iv) a = -1, d = ^{1}/2

(v) a = -1.25, d = -0.25

**3.** For the following APs, write the first term and the common difference.

**4.** Which of the following are APs? If they form an AP, find the common difference d and write three more terms.

**Exercise 2**

**5.** Find the missing variable from a, d, n and *a _{n}, *where a is the first term, d is the common difference and

*ar,*is the nth term of AP.

(i) a = 7, d = 3, n = 8 (ii) a = -18, n = 10, *a _{n} = 0*

(iii) d = -3, n = 18, *a _{n}= -5 *(iv) a = -18.9, d = 2.5,

*a*3.6

_{n}=(v) a = 3.5, d = 0, n = 105

**6.** Choose the correct choice in the following and justify:

(i) 30th term of the AP: 10, 7, 4…… is

(A) 97 (B) 77 (C) -77 (D) -87

(ii) 11th term of the AP: -3, -12, 2… is

**7. ** In the following AP’s find the missing terms:

(i) 2, __, 26 (ii) __, 13, __ 3

(iii) 5, __, __, 9^{11}. (iv) -4, __, __, __, __, 6

(v) __, 38, __, __, __, -22

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

**9. ** Find the number of terms in each of the following APs:

**10.**Check whether -150 is a term of the AP: 11, 8, 5, 2…

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

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

**13.** If the third and the ninth terms of an AP are 4 and -8 respectively, which term of this AP is zero?

**14.** The 17th term of an AP exceeds its 10th term by 7. Find the common difference.

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

**16.** Two AP’s have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms.

**17.**How many three digit numbers are divisible by 7?

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

**19**. For what value of n, are the nth terms of two AP’s : 63, 65, 67… and 3, 10, 17… equal?

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

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

**22**. The sum of the 4th and 8th terms of an AP is 24 and the sum of 6th and 10th terms is 44. Find the three terms of the AP.

**23**. Subba Rao started work in 1995 at an annual salary of Rs 5000 and received an increment of Rs 200 each year. In which year did his income reach Rs 7000?

**24**. Ramkali saved Rs. 5 in the first week of a year and then increased her weekly savings by Rs. 1.75. If in the nth week, her weekly savings become Rs 20.75, find n.

**Exercise 3**

**25.** Find the sum of the following AP’s.

(i) 2, 7, 12… to 10 terms (ii) -37, -33, -29… to 12 terms

(iii) 0.6, 1.7, 2.8… to 100 terms (iv)

**26.** Find the sums given below:

**27**. In an AP

(i) given *a = *5, *d = *3, *an = *50, find n and S_{n}.

(ii) given *a = *7, *a13 = *35, find d and S_{13}

(iii) given *a _{12} = *37,

*d =*3, find a and S

_{12}.

(iv) given *a _{3} = *15, S

_{10}= 125, find d and a

_{10}.

(v) given *d = *5, S_{9} = 75, find a and a_{9}.

(vi) given *a = *2, *d = *8, *S _{n}, = *90, find n and

*a*

_{n}(vii) given *a= *8, *a _{n}= *62,

*S*210, find n and d,

_{n}=(viii) given *a _{n}= *4,

*d =*2, S

_{n}= -14, find n and a.

(ix) given *a = *3, *n = *8, *S = *192, find d.

(x) given *I = *28, *S = *144, and there are total of 9 terms. Find a.

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

**29.** 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.

**30.** The first and the last terms of an AP are 17 and 350 respectively. If, the common difference is 9, how many terms are there and what is their sum?

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

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

**33.** 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.

**34.** Show that a_{1}, a_{2….}a_{n} form an AP where a_{n} is defined as below:

(i) a_{n} = 3 + 4n (ii) a_{n} = 9 – 5n

Also find the sum of the first 15 terms in each case.

**35.** If the sum of the first n terms of an AP is (4n – n^{2}), what is the first term (that is S_{1})? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

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

**37**. Find the sum of the first 15 multiples of 8.

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

**39**. 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?

**40**. 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 term, find the value of each of the prizes.

**41.** 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 two trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

**42.** A spiral is made up of successive semicircles, with centers alternatively at A and B, starting with center 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.

**43.** 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 many rows are the 200 logs placed and how many logs are in the top row?

**44.** In a potato race, a bucket is placed at the starting point, which is 5 meters from the first potato, and the other potatoes are placed 3 meters 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?

**Exercise 4**

**45.** Which term of the AP: 121, 117, 113, ……. is its first negative term?

**46.** The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of sixteen terms of the AP.

**47.** A ladder has rungs 25 cm apart (see figure). The rungs decrease uniformly in length from 45 cm, at the bottom to 25 cm at the top. If the top and the bottom rungs are m apart, what is the length of the wood required for the rungs?

**48.** 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.

**49.** A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete.

Each step has a rise of m and a tread of m (see figure). Calculate the total volume of concrete required to build the terrace.