5. Arithmetic progressions

5.1 Introduction

5.2 Arithmetic Progressions

Example 1:

For the AP : 3/2,1/2, – 1/2, – 3/2, . . ., write the first term a and the common difference d.

Example 2:

Which of the following list of numbers form an AP? If they form an AP, write the next two terms :

(i) 4, 10, 16, 22, . . . (ii) 1, – 1, – 3, – 5, . . .

(iii) – 2, 2, – 2, 2, – 2, . . . (iv) 1, 1, 1, 2, 2, 2, 3, 3, 3, . . .

Exercise 5.1

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 ₹ 15 for the first km and ₹ 8 for each additional km.

(ii) 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.

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

(iv) The amount of money in the account every year, when ₹ 10000 is deposited at compound interest at 8 % per annum.

2. Write first four terms of the AP, when the first term a and the 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:

(i) 3, 1, – 1, – 3, . . . (ii) – 5, – 1, 3, 7, . . .

(iii) 1 , 5 , 9 , 13 ,3 3 3 3. . . (iv) 0.6, 1.7, 2.8, 3.9, . . .

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

(i) 2, 4, 8, 16, . . . (ii) 2, 5/2 , 3, 7/2 . . . (iii) – 1.2, – 3.2, – 5.2, – 7.2, . . .

(iv) – 10, – 6, – 2, 2, . . . (v) 3, 3 + √2 , 3 + 2√2 , 3 + 3√2 , . . . (vi) 0.2, 0.22, 0.222, 0.2222, . . .

(vii) 0, – 4, – 8, –12, . . . (viii) –1/2, –1/2, –1/2, –1/2, . . .

(ix) 1, 3, 9, 27, . . . (x) a, 2a, 3a, 4a, . . .

(xi) a, a², a³, a⁴, . . . (xii) √2, √8, √18 , √32, . . .

(xiii) √3, √6, √9 , √12 , . . . (xiv) 1², 3², 5², 7², . . .

(xv) 1², 5², 7², 73, . . .

5.3 nth Term of an AP

Example 3:

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

Example 4:

Which term of the AP : 21, 18, 15, . . . is – 81? Also, is any term 0? Give reason for your answer.

Example 5:

Determine the AP whose 3rd term is 5 and the 7th term is 9.

Example 6:

Check whether 301 is a term of the list of numbers 5, 11, 17, 23, . . .

Example 7:

How many two-digit numbers are divisible by 3?

Example 8:

Find the 11th term from the last term (towards the first term) of the AP : 10, 7, 4, . . ., – 62.

Example 9:

A sum of ₹ 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.

Example 10:

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?

EXERCISE 5.2

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

	a	d	n	a_n
(i)	7	3	8	...
(ii)	-18	...	10	0
(iii)	...	-3	18	-5
(iv)	-18.9	2.5	...	3.6
(v)	3.5	0	105	...

2. 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,-1/2 , 2, . . ., is

(A) 28 (B) 22 (C) –38 (D) – 48 1/2

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

(i) 2, □, 26

(ii) □, 13, □, 3

(iii) 5, □ , □,9 1/2

(iv) – 4, □, □, □, □, 6

(v) □, 38, □, □, □, – 22

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

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

(i) 7, 13, 19, . . . , 205 (ii) 18, 15 1/2, 13, . . . , – 47

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

5.4 Sum of First n Terms of an AP

Example 11:

Find the sum of the first 22 terms of the AP : 8, 3, –2, . . .

Example 12:

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

Example 13:

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

Example 14:

Find the sum of :

(i) the first 1000 positive integers (ii) the first n positive integers

Example 15:

Find the sum of first 24 terms of the list of numbers whose nth term is given by

a_n = 3 + 2n

Example 16:

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

Exercise 5.3

1. Find the sum of the following APs:

(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) 1/15, 1/12 , 1/10 , . . ., to 11 terms.

2. Find the sums given below :

(i) 7 + 10 1/2 + 14 + . . . + 84 (ii) 34 + 32 + 30 + . . . + 10

(iii) –5 + (–8) + (–11) + . . . + (–230)

3. In an AP:

(i) given a = 5, d = 3, a_n = 50, find n and S_n.

(ii) given a = 7, a₁₃ = 35, find d and S₁₃.

(iii) given a₁₂ = 37, d = 3, find a and S₁₂

(iv) given a₃ = 15, S₁₀ = 125, find d and a₁₀.

(v) given d = 5, S₉ = 75, find a and a₉.

(vi) given a = 2, d = 8, S_n = 90, find n and a_n.

(vii) given a = 8, a_n = 62, S_n = 210, find n and d.

(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 l = 28, S = 144, and there are total 9 terms. Find a.

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

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.

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?

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

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

9. 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.

10. Show that a₁, a₂, . . ., a_n, . . . form an AP where an 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.

11. If the sum of the first n terms of an AP is 4n – 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.

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

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

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

15. A contract on 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?

16. A sum of ₹ 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹ 20 less than its preceding prize, find the value of each of the prizes.

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 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?

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. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take ℼ = 22/7 )

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. 5.5). In how many rows are the 200 logs placed and how many logs are in the top row?

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 3 m apart in a straight line. There are ten potatoes in the line (see Fig. 5.6).

Exercise 5.4

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

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.

3. A ladder has rungs 25 cm apart. (see Fig. 5.7). 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?

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.

5. 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 1/4m and a tread of 1/2m. (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace.