NCERT Solutions for class 10th Maths Chapter 5 Arithmetic Progressions

Exercise 5.1

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

  1. The taxi fare after each km when the fare is < 15 for the first km and < 8 for each additional km.
  2. The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air remaining in the cylinder at a time.
  3. The cost of digging a well after every metre of digging, when it casts rs.150 for the first metre and rises by rs.50 for each subsequent metre.
  4. The amount of money in the account every year, when rs.10000 is deposited at compound interest at 8% per annum.

Sol. (i) tn denotes the taxi fare (in rs.) for the first n km.
Now, t1 = 15, t2 = 15 + 8 = 23,
t3 = 23 + 8 = 31, t4 = 31+ 8 = 39, …..
List of fares after 1km, 2km, 3km, 4km,… respectively is 15, 23, 31, 39, (in<)
Here, the difference between the consecutive terms is constant, i.e; t2-t1=t3-t2=t4-t3 =….= 8

Note: We can also write the list of amounts for each year as :
10000, 10000 x (1+8/100), 10000x(1+8/100)2, 10000 x (1+8/100)3…..

Question 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

Note: a, a+d, a+2d, a+3d,represents an arithmetic progression where a is the first term and d the common difference.This is called the general form of an AP

Question 3. For the following APs, write the first term and the common difference:
(i) 3, 1, -1, -3, ….. (ii) -5, -1, 3, 7, …..
(iii) 1/3,5/3,9/3,13/3····· (iv) 0.6, 1.7, 2.8, 3.9, …..

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

Exercise 5.2

Question 1. Fill in the blanks in the following table, given that a is the first term, d the common difference an and the nth term of the AP:

Question 2. Choose the correct choice in the following and justify:

Question 3. In the following APs, find the missing terms in the boxes:

Question 4. Which term of the AP: 3, 8, 13, 18,is 78?

Question 5. Find the number of terms in each of the following APs:

Question 6. Check whether-150 is a term of the AP: 11, 8, 5, 2….

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

Question 8. An AP consists of 5O terms of which 3rd term is 12 and the last term is 106.Find the 29th term.

Question 9. If the 3rd and the 9th term ofanAP are 4 and -8, respectively.Which term of this AP is zero?

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

Question 11. Which term of the A.P. 3, 15, 27, 39,will be 132 more than its 54th term?

Question 12. Two APs have the same common difference. The difference between their 100th terms is 100,what is the difference between their 1000th terms?.

Question 13. How many three-digit numbers are divisible by 7.

Question 14. How many multiples of 4 lie between 10 and 250?.

Question 15. For what value of n, are the nth terms of two APs: 63,65,67,and 3,10,17,….equal?

Question 16. Determine the A.P. whose third term is 16 and the difference of 5th term from 7th term is 12.

Question 17. Find the 20th term from the last term of the AP: 3, 8, 13, 253,

Note: nth terms from the last can also be calculated using formula given below:
an from last= l-(n-l)d, where, l=last term

Question 18. The sum of the 4th and 8th terms of an AP is 24 and the some of the 6th and 10 terms is 44. Find the first three terms of the AP.

Question 19. 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?

Question 20. Ramkali saved rs.5 in the first week of a year and then increased her weekly saving by rs.1.75.If in the nth week, her weekly savings become rs.20.75, find n.

Exercise 5.3

Question 1. Find the sum of the following APs:

Question 2. Find the sums given below:

Note: In an AP, when a, l, n is given, Sum of first n terms can also be calculated using formula. Sn=n/2(a+l)

Question 3. In an AP:

Note: We also use S in place of Sn to denote the sum of first n terms of the AP

Question 4. How many terms of the AP: 9, 17, 25,must be taken to give sum of 636?

Question 5. 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.

Question 6. 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?

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

Question 8. Find the sum of the first 51 terms of the A.P. whose second and third term are respectively 14 and 18.

Question 9. If the sum of 7 terms of an AP is 49 and thatof17 terms is 289, find the sum
of n terms

Question 10. Show that a1+ a2,….., a11, •••••• form an AP where a11 is defined as below :
(i) a = 3 + 4n (ii)a = 9 – 5
Also, find the sum of the first 15 terms in each case.

Question 11. If the sum of the first n terms of an AP is 4 -n2, what is the first term (that is S1)? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

Question 12. Find the sum of the first 40 positive integers divisible by 6.

Question 13. Find the sum of the first 15 multiples of 8.

Question 14. Find the sum of the odd numbers between 0 and 50.

Question 15. A contract on a construction job specifies a penalty for delay of completion beyond a certain date as follows: 200 for the first day, 250 for the second day, 300 for the third day, etc., the penalty for each succeeding day being 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?

Question 16. A sum of rs700 is to be used to given seven cash prizes to students of a school for their overall academic performance. If each prize is rs 20 less than its proceeding prize, find the value of each of the prizes.

Question 17. 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 one 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?

Question 18. 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,… as shown in Fig. What is the total length of such a spiral made up of thirteen consecutive semicircles? (take pi=22/7)
[Hint : Length of successive semicircles is 11, 12, 13, 14,with centres at A, B, A, B, respectively.]

Question 19. 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 (see Fig.). In how many rows are the 200 logs placed and how many logs are in the top row?

Question 20. 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 3m apart in a straight line. There are ten potatoes in the line (see Fig.)

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? [Hint: To pickup the first potato and the second potato, the total distance (in metres) run by a competitor is 2 x 5 + 2 x (5 + 3)]

Exercise 5.4

Question 1. Which term of the AP: 121, 177, 113,…,is its first negative term?

Hint: (Find n for an < 0)

Question 2. The sum of the third and the seventh terms of an AP is 6 and their product is 8.Find the sum of first sixteen terms of the AP

Question 3. A ladder has rungs 25 cm apart. (see fig.).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 2 1/2 m apart, what is the length of the wood required for the rungs?
(Hint: Number of rungs = 250/25)

Question 4. 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.

Question 5. A small terrace at a football ground comprises of15 steps each of which is 50 m long and built of solid concrete.
Each step has a rise of 1/4 m and a tread of 1/2 m.(see Fig).Calculate the total volume of concrete required to build the terrace.
[Hint :
Volume of concrete required to build the first step = 1/4 x 1/2 x 50m3]

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