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Arithmetic Progressions — Class 10 Mathematics

Mathematics · Chapter 5

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Chapter summary

An arithmetic progression (AP) is a list of numbers in which each term is obtained by adding a fixed number, the common difference, to the previous term. The chapter derives two central formulae: the nth term , and the sum of the first terms (equivalently when the last term is known). These let students find any term, count terms, and add long sequences quickly. APs model many real situations such as savings, salaries and seating arrangements.

nth term of an APSum of first n termsApplications of AP

Key ideas & terms

Arithmetic progression: A sequence with a constant difference between consecutive terms.
Common difference (d): The fixed amount added to get the next term.
nth term: .
Sum of n terms: .

Extra questions & answers

Find the 10th term of the AP: 2, 5, 8, …

, so .

Write the formula for the sum of the first terms of an AP.

, or where is the last term.

What is an arithmetic progression (AP)?

A list of numbers in which each term is obtained by adding a fixed number, the common difference d, to the preceding term.

Write the formula for the nth term of an AP.

, where is the first term and is the common difference.

Write the formula for the sum of the first n terms of an AP.

, or when the last term is known.

Find the common difference and the 10th term of the AP 3, 7, 11, …

; .

How many two-digit numbers are divisible by 3?

They form the AP 12, 15, …, 99; with : numbers.

Find the sum of the first 20 terms of the AP 2, 5, 8, …

: .

Which term of the AP 5, 9, 13, … is 81?

; so 81 is the 20th term.

Is 0 a term of the AP 31, 28, 25, …? Explain.

: gives , which is not a whole number, so 0 is not a term.

EXERCISE 5.1

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

The fares after successive kilometres are . The difference between consecutive fares is always , so the list forms an AP with common difference .

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

Let the initial amount of air be . The successive amounts are . Their consecutive differences are not equal, so the list does not form an AP.

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

The costs are . Each cost is more than the preceding one, so the list forms an AP with .

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

The amounts are . The increases are , which are not equal. Therefore, the list does not form an AP.

2. (i)

The first four terms are , so they are .

2. (ii)

The first four terms are .

2. (iii)

The first four terms are .

2. (iv)

The first four terms are .

2. (v)

The first four terms are .

3. (i)

The first term is , and the common difference is .

3. (ii)

The first term is , and the common difference is .

3. (iii)

The first term is , and the common difference is .

3. (iv)

The first term is , and the common difference is .

4. (i)

This is not an AP because the consecutive differences are , which are not equal.

4. (ii)

This is an AP with . The next three terms are .

4. (iii)

This is an AP with . The next three terms are .

4. (iv)

This is an AP with . The next three terms are .

4. (v)

This is an AP with . The next three terms are .

4. (vi)

This is not an AP because its consecutive differences are not equal.

4. (vii)

This is an AP with . The next three terms are .

4. (viii)

This is an AP with . The next three terms are .

4. (ix)

This is not an AP because the consecutive differences are , which are not equal.

4. (x)

This is an AP with common difference . The next three terms are .

4. (xi)

In general, this is not an AP because the differences are not equal.

4. (xii)

Writing the terms as , we see that this is an AP with . The next three terms are , that is, .

4. (xiii)

This is not an AP because its consecutive differences are not equal.

4. (xiv)

The terms are , whose consecutive differences are . Hence, it is not an AP.

4. (xv)

The terms are . Their consecutive differences are all , so this is an AP with . The next three terms are .

EXERCISE 5.2

1. (i)

Using , .

1. (ii)

From , we obtain .

1. (iii)

From , we obtain .

1. (iv)

From , we get , so .

1. (v)

Since , every term is . Therefore, .

2. (i) th term of the AP: , is (A) (B) (C) (D)

Here and . Thus, . The correct choice is (C) .

2. (ii) th term of the AP: , is (A) (B) (C) (D)

Here and . Therefore, . The correct choice is (B) .

3. (i)

There are three terms, so the middle term is the arithmetic mean: . The missing term is .

3. (ii)

Let the first term be and common difference be . Since and , , so and . The completed AP is .

3. (iii)

Here , giving . The completed AP is .

3. (iv)

There are six terms, so . The completed AP is .

3. (v)

Let the first term be . Since and , , so and . The completed AP is .

4. Which term of the AP : , is ?

Here and . Solving gives , so . Thus, is the th term.

5. (i)

Here , , and . Thus, , giving . There are terms.

5. (ii)

Here and . Solving gives , so .

6. Check whether is a term of the AP :

Here and . If were the th term, then , giving . Since this is not a positive integer, is not a term of the AP.

7. Find the st term of an AP whose th term is and the th term is .

We have and . Subtracting gives , so , and then . Therefore, .

8. An AP consists of terms of which rd term is and the last term is . Find the th term.

We have and . Hence, , so and . Therefore, .

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

The equations are and . Thus, , so and . Setting , , which gives . The th term is zero.

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

Since , we have . Therefore, .

11. Which term of the AP : will be more than its th term?

The common difference is . If the th term is more than the th term, then . Thus, , so .

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

Let their first terms be and , with common difference . The difference between corresponding terms is always . Therefore, the difference between their th terms is also .

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

The first three-digit multiple of is , and the last is . These form an AP with . From , we get . Thus, there are such numbers.

14. How many multiples of lie between and ?

The multiples are . Thus, , giving . Hence, multiples of lie between and .

15. For what value of , are the th terms of two APs: and equal?

The th terms are and . Equating them gives , so . Therefore, .

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

We have . Also, , so . Hence, , and the AP is .

17. Find the th term from the last term of the AP : .

First, , so the AP has terms. The th term from the end is the nd term. Hence, .

18. The sum of the th and th terms of an AP is and the sum of the th and th terms is . Find the first three terms of the AP.

We have and . Subtracting gives , so . Then , giving . The first three terms are .

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

The annual salaries form an AP with and . Solving gives . The th year after starting in is .

20. Ramkali saved in the first week of a year and then increased her weekly savings by . If in the th week, her weekly savings become , find .

Using , . Thus, , so .

EXERCISE 5.3

1. (i) , to terms.

Here , , and . Therefore, .

1. (ii) , to terms.

Here , , and . Thus, .

1. (iii) , to terms.

Here , , and . Hence, .

1. (iv) , to terms.

The common difference is . Therefore, .

2. (i)

This is an AP with , , and last term . Solving gives . Thus, the sum is .

2. (ii)

Here , , and . There are terms, so .

2. (iii)

Here , , and . Solving gives . Hence, .

3. (i) given , find and .

From , we get . Therefore, .

3. (ii) given , find and .

From , . Hence, .

3. (iii) given , find and .

From , we get . Thus, .

3. (iv) given , find and .

The equations are and . Thus, and , giving and . Therefore, .

3. (v) given , find and .

Using , we obtain . Hence, .

3. (vi) given , find and .

The equation simplifies to . Its positive solution is . Therefore, .

3. (vii) given , find and .

Using , , so . Then , giving .

3. (viii) given , find and .

Since , . Also, . Thus, , and .

3. (ix) given , find .

Using , we get , so .

3. (x) given , and there are total terms. Find .

Using , . Therefore, , so .

4. How many terms of the AP : must be taken to give a sum of ?

Here and . The equation becomes . Its positive solution is .

5. The first term of an AP is , the last term is and the sum is . Find the number of terms and the common difference.

Using , we get . Then , so .

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

From , we obtain . Therefore, .

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

Since , the first term is . Thus, .

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

The common difference is , and . Therefore, .

9. If the sum of first terms of an AP is and that of terms is , find the sum of first terms.

From , we get . From , we get . Hence, and . Therefore, .

10. (i) Show that form an AP where . Also find the sum of the first terms.

We have , which is constant, so the terms form an AP. Here , , and .

10. (ii) Show that form an AP where . Also find the sum of the first terms.

We have , which is constant, so the terms form an AP. Here , , and .

11. If the sum of the first terms of an AP is , what is the first term (that is )? What is the sum of first two terms? What is the second term? Similarly, find the rd, the th and the th terms.

Since , , so the first term is . Also, , so the second term is . Further, , , and in general .

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

The numbers are . Their sum is .

13. Find the sum of the first multiples of .

The sum is .

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

The odd numbers are . There are terms, and their sum is .

15. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: for the first day, for the second day, for the third day, etc., the penalty for each succeeding day being more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by days?

The daily penalties form an AP with , , and . Therefore, .

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

Let the first prize be . Here , , and . Thus, , giving . The prizes are , and .

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 tree, a section of Class II will plant trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

One section from Classes I to XII plants trees. Since each class has three sections, the total is trees.

18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii as shown in Fig. . What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take )

The length of a semicircle of radius is . The radii form an AP with , , and . Their sum is . Therefore, the spiral length is .

19. logs are stacked in the following manner: logs in the bottom row, in the next row, in the row next to it and so on (see Fig. ). In how many rows are the logs placed and how many logs are in the top row?

If there are rows, the top row has logs. Hence, , giving . The valid solution is , so the top row has logs.

20. In a potato race, a bucket is placed at the starting point, which is from the first potato, and the other potatoes are placed 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?

The one-way distances are . Their sum is . Each trip is made in both directions, so the total distance is .

EXERCISE 5.4 (Optional)*

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

Here and , so . For the first negative term, , giving . Therefore, the nd term is the first negative term; it equals .

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

Let the first term be and common difference be . From , , so . Also, , or . Thus, . If , then and . If , then and . Hence, the possible sums are and .

3. A ladder has rungs apart. (see Fig. ). The rungs decrease uniformly in length from at the bottom to at the top. If the top and the bottom rungs are apart, what is the length of the wood required for the rungs?

The distance between the top and bottom rungs is . Therefore, the number of rungs is . Their lengths form an AP from to . Thus, the required length is , or .

4. The houses of a row are numbered consecutively from to . Show that there is a value of such that the sum of the numbers of the houses preceding the house numbered is equal to the sum of the numbers of the houses following it. Find this value of .

The sum before house is . The total of all house numbers is , so the sum after house is . Equating the two gives , so .

5. A small terrace at a football ground comprises of steps each of which is long and built of solid concrete. Each step has a rise of and a tread of . (see Fig. ). Calculate the total volume of concrete required to build the terrace.

The volumes corresponding to the successive step heights form an AP. The total volume is . Since , the required volume is .