#### Chapters

Chapter 2: Polynomials

Chapter 3: Pair of Linear Equations in Two Variables

Chapter 4: Quadratic Equations

Chapter 5: Arithmetic Progressions

Chapter 6: Triangles

Chapter 7: Coordinate Geometry

Chapter 8: Introduction to Trigonometry

Chapter 9: Some Applications of Trigonometry

Chapter 10: Circles

Chapter 11: Constructions

Chapter 12: Areas Related to Circles

Chapter 13: Surface Areas and Volumes

Chapter 14: Statistics

Chapter 15: Probability

#### NCERT Mathematics Class 10

## Chapter 5 : Arithmetic Progressions

#### Pages 99 - 100

In which of the following situations, does the list of numbers involved make as arithmetic progression and why?

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

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

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.

In which of the following situations, does the list of numbers involved make as arithmetic progression and why?

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

In which of the following situations, does the list of numbers involved make as arithmetic progression and why?

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

Write first four terms of the A.P. when the first term a and the common differenced are given as follows *a* = 10, *d* = 10

Write first four terms of the A.P. when the first term a and the common differenced are given as follows *a* = -2, *d* = 0

Write first four terms of the A.P. when the first term a and the common differenced are given as follows *a* = 4, *d* = - 3

Write first four terms of the A.P. when the first term a and the common differenced are given as follows `a = -1, d = 1/2`

Write first four terms of the A.P. when the first term a and the common differenced are given as follows a = - 1.25, d = - 0.25

For the following APs, write the first term and the common difference 3, 1, – 1, – 3, . . .

For the following A.P.s, write the first term and the common difference -5, - 1, 3, 7

For the following A.P.s, write the first term and the common difference `1/3, 5/3, 9/3, 13/3` ....

For the following A.P.s, write the first term and the common difference. 0.6, 1.7, 2.8, 3.9

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms 2, 4, 8, 16 …

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms `2, 5/2, 3, 7/2 ....`

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms. -1.2, -3.2, -5.2, -7.2 …

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms. -10, - 6, - 2, 2 …

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms. `3, 3 + sqrt2, 3 + 2sqrt2, 3 + 3sqrt2`

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms. 0.2, 0.22, 0.222, 0.2222 ….

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms. 0, - 4, - 8, - 12 …

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms. -1/2, -1/2, -1/2, -1/2 ....

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms 1, 3, 9, 27 …

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms. *a*, 2*a*, 3*a*, 4*a* …

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms a, a^{2}, a^{3}, a^{4} …

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms. `sqrt2, sqrt8, sqrt18, sqrt32 ...`

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms `sqrt3, sqrt6, sqrt9, sqrt12 ...`

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms. 1^{2}, 3^{2}, 5^{2}, 7^{2} …

Which of the following are APs? If they form an A.P. find the common difference *d* and write three more terms. 1^{2}, 5^{2}, 7^{2}, 7^{3} …

#### Pages 105 - 107

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:

Choose the correct choice in the following and justify 30th term of the AP: 10, 7, 4, . . . , is

(A) 97

(B) 77

(C) –77

(D) – 87

Choose the correct choice in the following and justify

11^{th }term of the A.P. `-3, -1/2, 2` .... is

(A) 28

(B) 22

(C) –38

(D) – 48`1/2`

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

In the following APs find the missing term in the boxes

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

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

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

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

Find the number of terms in the following A.P. : 7, 13, 19, . . . , 205

Find the number of terms in each of the following A.P. `18,15 1/2, 13, . . . , – 47`

Find the 31^{st} term of an A.P. whose 11^{th} term is 38 and the 16^{th} term is 73

An A.P. consists of 50 terms of which 3^{rd} term is 12 and the last term is 106. Find the 29^{th} term

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

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

Which term of the A.P. 3, 15, 27, 39, … will be 132 more than its 54^{th}term?

Two APs have the same common difference. The difference between their 100^{th} term is 100, what is the difference between their 1000^{th}terms?

How many three digit numbers are divisible by 7

How many multiples of 4 lie between 10 and 250?

For what value of *n*, are the n^{th} terms of two APs 63, 65, 67, and 3, 10, 17, … equal?

Determine the A.P. whose third term is 16 and the 7^{th} term exceeds the 5^{th} term by 12.

Find the 20^{th} term from the last term of the A.P. 3, 8, 13, …, 253

The sum of 4^{th} and 8^{th} terms of an A.P. is 24 and the sum of the 6^{th} and 10^{th} terms is 44. Find the first three terms of the A.P.

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?

Ramkali saved Rs 5 in the first week of a year and then increased her weekly saving by Rs 1.75. If in the *n*^{th}week, her week, her weekly savings become Rs 20.75, find *n.*

#### Pages 112 - 114

Find the sum of the following APs:

2, 7, 12, . . ., to 10 terms.

Find the sum of the following APs.

− 37, − 33, − 29 ,…, to 12 terms

Find the sum of the following APs.

0.6, 1.7, 2.8 ,…….., to 100 terms

Find the sum of the following APs. `1/15, 1/12, 1/10` , ...... , to 11 terms

Find the sums given below : 7 + `10 1/2` + 14 + .................. +84

Find the sums given below :

34 + 32 + 30 + . . . + 10

Find the sums given below :

–5 + (–8) + (–11) + . . . + (–230)

In an AP:

Given a = 5, d = 3, a_{n} = 50, find n and S_{n}.

In an AP:

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

In an AP

Given *a* = 7, *a*_{13} = 35, find *d* and *S*_{13}.

In an AP

Given a_{12} = 37, d = 3, find a and S_{12}.

In an AP

Given a_{3} = 15, S_{10} = 125, find d and a_{10}.

In an AP

Given *d* = 5, *S*_{9} = 75, find *a* and *a*_{9}.

In an AP:

Given *a* = 2, *d* = 8, *S*_{n} = 90, find *n *and *a*_{n}.

In an AP

Given a = 8, a_{n} = 62, S_{n} = 210, find n and d.

In an AP .Given a_{n} = 4, d = 2, S_{n} = − 14, find n and a.

In an AP

Given a = 3, n = 8, S = 192, find d.

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

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.

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

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

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

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

a_{n} = 3 + 4n

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

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

a_{n} = 9 − 5n

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

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 3^{rd}, the10^{th} and the n^{th} terms.

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

Find the sum of first 15 multiples of 8.

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

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.

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.

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?

A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, ……… as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take `pi = 22/7` )

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?

In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato and 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?

[Hint: to pick up the first potato and the second potato, the total distance (in metres) run by a competitor is 2 × 5 + 2 ×(5 + 3)]

#### Page 115

Which term of the A.P. 121, 117, 113 … is its first negative term? [Hint: Find *n* for *a*_{n} < 0]

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.

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 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 ]

The houses in a row numbered consecutively from 1 to 49. Show that there exists a value of X such that sum of numbers of houses preceding the house numbered X is equal to sum of the numbers of houses following X.

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 1/4 m and a tread of 1/2 m (See figure) 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 50m^{3}]

#### Extra questions

Find the sum of all natural numbers between 250 and 1000 which are exactly divisible by 3

If five times the fifth term of an A.P. is equal to 8 times its eight term, show that its 13^{th} term is zero

If the nth term of an AP is (2n + 1) then find the sum of its first three terms

A. 6*n* + 3

B. 15

C. 12

D. 21

The sum of n, 2n, 3n terms of an A.P. are S_{1} , S_{2} , S_{3} respectively. Prove that S_{3} = 3(S_{2} – S_{1} )

The sum of three numbers in A.P. is –3, and their product is 8. Find the numbers

If the term of m terms of an A.P. is the same as the sum of its n terms, show that the sum of its (m + n) terms is zero

Find the sum of 20 terms of the A.P. 1, 4, 7, 10, ……

The ratio of the sums of m and n terms of an A.P. is m^{2} : n^{2}. Show that the ratio of the m^{th} and n^{th} terms is (2m – 1) : (2n – 1)

In an AP

Given *a* = 2, *d* = 8, *S*_{n} = 90, find *n *and *a*_{n}.

Find the sum of first 30 terms of an A.P. whose second term is 2 and seventh term is 22

The ratio of the sum use of n terms of two A.P.’s is (7n + 1) : (4n + 27). Find the ratio of their m^{th} terms

Divide 32 into four parts which are in A.P. such that the product of extremes is to the product of means is 7 : 15.

If the m^{th} term of an A.P. is 1/n and the n^{th} term is 1/m, show that the sum of mn terms is (mn + 1)

The sum of the first p, q, r terms of an A.P. are a, b, c respectively. Show that `\frac { a }{ p } (q – r) + \frac { b }{ q } (r – p) + \frac { c }{ r } (p – q) = 0`

The sum of n terms of three arithmetical progression are S_{1} , S_{2} and S_{3} . The first term of each is unity and the common differences are 1, 2 and 3 respectively. Prove that S_{1} + S_{3} = 2S_{2}

How many terms of the series 54, 51, 48, …. be taken so that their sum is 513 ? Explain the double answer

Determine the 10th term from the end of the A.P. 4, 9, 14, …….., 254

The 10^{th} term of an A.P. is 52 and 16^{th} term is 82. Find the 32nd term and the general term

If p^{th} , q^{th} and r^{th} terms of an A.P. are a, b, c respectively, then show that

(i) a (q – r) + b(r – p) + c(p – q) = 0

(ii) (a – b) r + (b –¬ c) p + (c – a) q = 0

Which term of the sequence –1, 3, 7, 11, ….. , is 95 ?

Four numbers are in A.P. If their sum is 20 and the sum of their square is 120, then find the middle terms

If the pth term of an A.P. is q and the qth term is p, prove that its nth term is (p + q – n)

Find the common difference of an AP, whose first term is 5 and the sum of its first four terms is half the sum of the next four terms

Write the first five terms of the sequence defined by `a_n = (–1)^(n-1) . 2^n`

Write the first three terms in the sequences defined by the following

a_{n} = n^{2} + 1

Determine the general term of an A.P. whose 7^{th} term is –1 and 16^{th} term 17

Find 26^{th} term from last of an AP 7, 15, 23……., 767 consits 96 terms

The nth term of a sequence is 3n – 2. Is the sequence an A.P. ? If so, find its 10^{th} term

Write the first three terms in each of the sequences defined by the following

a_{n} = 3n + 2

Find the 12^{th} , 24^{th} and n^{th} term of the A.P. given by 9, 13, 17, 21, 25, ………

Find four numbers in A.P. whose sum is 20 and the sum of whose squares is 120

If the m^{th} term of an A.P. be `1/n` and n^{th} term be `1/m`, then show that its (mn)^{th} term is 1.

If m times m^{th} term of an A.P. is equal to n times its n^{th} term, show that the (m + n) term of the A.P. is zero

Which term of the sequence 4, 9 , 14, 19, …… is 124 ?

If the n^{th} term of a progression be a linear expression in n, then prove that this progression is an AP

#### NCERT Mathematics Class 10

#### Textbook solutions for Class 10

## NCERT solutions for Class 10 Mathematics chapter 5 - Arithmetic Progressions

NCERT solutions for Class 10 Maths chapter 5 (Arithmetic Progressions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Textbook for Class 10 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10 Mathematics chapter 5 Arithmetic Progressions are General Term of an Arithmetic Progression, nth Term of an AP, Sum of First n Terms of an AP, Arithmetic Progression, Derivation of the n th Term, Application in Solving Daily Life Problems, Arithmetic Progressions Examples and Solutions.

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