NCERT Solutions Class 10 Maths Chapter 5
NCERT Solutions For Class 10 Mathematics Chapter 5 Arithmetic Progressions
Mathematics is one subject that most students find difficult. The only way to actually score good marks in Mathematics is to practice solving as many problems as one can. That is why students find NCERT Solutions For Class 10 Mathematics Chapter 5 by Extramarks really useful in their preparation. If students ever get stuck on any of the exercise questions, they can always refer to these solutions. These are also great resources for completing assignments and for last-minute revision.
Access NCERT Solutions for Mathematics Chapter 5 – Arithmetic Progression
Chapter 5 of Class 10 Mathematics NCERT textbook introduces students to Arithmetic Progressions, which are nothing but lists of numbers where each term, except the first term, can be derived by adding or subtracting a fixed number to its preceding term. These are really useful to express patterns that we see in our daily lives and in nature. The chapter further discusses the different types of Arithmetic Progressions, the sum of Arithmetic Progressions, deriving a general formula for the nth term of an Arithmetic Progression, and much more.
What is Arithmetic Progression?
An Arithmetic Progression is a series of numbers where each number can be derived from its predecessor by adding or subtracting a fixed number. For example, consider the series of numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Here, each term can be derived from its predecessor by adding 1 to it.
The fixed difference between the terms of an AP is called the common difference. In the above example, the common difference is 1.
NCERT Solutions for Class 10 Mathematics
Extramarks provides detailed NCERT Solutions for Class 10 Mathematics. Since the only way to get good at Mathematics is to solve problems, these solutions can be very useful for students. Extramarks provides detailed solutions to the questions given in all of the NCERT Mathematics textbooks chapters listed below:
- Chapter 1 – Real Numbers
- 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
The General Form of an Arithmetic Progression
The general form of an AP can be expressed as follows:
A, A + d, A + 2d, A + 3d,….
Here, A is the first term and d is the common difference.
|Position Of Terms||Representation of terms||Value of Terms|
There are some essential formulae that students will need to solve problems related to Arithmetic Progressions. These formulae have been listed below:
- Arithmetic Progression’s nth Term (AP)
Students will face many problems related to this chapter where they will be expected to calculate the nth term of an Arithmetic Progression. The formula to calculate the nth term of an AP is shown below:
An = a + (n – 1) d
a= The initial term
d= The difference value
n= the number of terms
An= the nth term
Consider the following problem. Calculate the 15th term in the AP: 1, 2, 3, 4, 5 …
Here, we are supposed to calculate the 15th term. So, the value of n is 15. The first term, a, is 1 and the common difference is also 1. Using the formula to calculate the nth term of an AP, the 15th term can be calculated as follows:
A15 = 1 + (15 – 1) 1 = 1 + 14 = 15.
- The Sum of the First n Terms
Another common type of problem that students will face related to Arithmetic Progressions is to calculate the sum of the first n terms of a given AP. The formula to do so is shown below:
S = (n / 2)*(2a+(n−1)d)
n is the number of terms
a is the first term in the AP
d is the common difference
Here, using the formula the nth term of an AP, we can further simplify this expression to the one shown below:
S = (n/2)*(a+An)/2
An is the nth term of the AP
Also, if the nth term is actually the last term of the AP, then the formula is further simplified to the following:
S = n / 2 (first term + last term)
We’ve also included all of the crucial formulas in this chapter in a table for easy reference.
|General Form of AP||A, a + d, a + 2d, a + 3d, a + 4d, …, a + nd|
|The nth term of AP||An = a + (n – 1) x d|
|Sum of n terms in AP||S= (n/2)*(2a + (n – 1)d)|
|Sum of all terms in a finite AP with the last term as I||(n/2)*(a + I)|
Why should students refer to NCERT Solutions from Extramarks?
Extramarks is committed to supporting students in every way. . We work to ensure that students can manage their academic and personal responsibilities more efficiently. The NCERT Solutions provides solutions for the Class 10 Chapter by Extramarks will help students cross-check their answers or even get the right answers to the textbook questions. Additionally, students will also get an idea of how to attempt a question in the right manner.
The Benefits of Referring to NCERT Solutions Mathematics
NCERT Solutions for Class 10 Mathematics Chapter 5 from Extramarks is a useful resource that will definitely help students to step up their preparation. Here are some of the benefits:
- All the answers are written following the CBSE guidelines. When students study from it, they will get an edge over their peers.
- Subject matter experts write these solutions giving utmost importance to accuracy so that students are able to understand every concept and answer any question easily.
- All the answers are written in a step-by-step manner, ensuring students do not have any difficulty understanding how the answer is derived. This encourages the students to master the topic and increases their confidence in achieving a higher grade
Q.4 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:
Q.7 Which term of the AP: 3, 8, 13, 18, . . . , is 78?
Q.9 Check whether –150 is a term of the AP: 11, 8, 5, 2, . . .
Q.10 Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.
Q.11 An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.
Q.12 If the 3rd and the 9th terms of an AP are 4 and –8 respectively, which term of this AP is zero?
Q.13 The 17th term of an AP exceeds its 10th term by 7. Find the common difference.
Q.14 Which term of the AP: 3, 15, 27, 39, . . . will be 132 more than its 54th term?
Q.15 Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?
Q.16 How many three-digit numbers are divisible by 7?
Q.17 How many multiples of 4 lie between 10 and 250?
Q.18 For what value of n, are the nth terms of two APs: 63, 65, 67, . . . and 3, 10, 17, . . . equal?
Q.19 Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.
Q.20 Find the 20th term from the last term of the AP: 3, 8, 13, . . ., 253.
Q.21 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.
Q.22 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?
Q.23 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.
Q.28 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.
Q.29 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?
Q.30 Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.
Q.31 Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
Q.32 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.
Q.33 Show that a1, a2, . . ., an, . . . form an AP where an is defined as below :
(i) an = 3 + 4n (ii) an = 9 – 5n
Also find the sum of the first 15 terms in each case.
Q.34 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.
Q.35 Find the sum of the first 40 positive integers divisible by 6.
Q.36 Find the sum of the first 15 multiples of 8.
Q.37 Find the sum of the odd numbers between 0 and 50.
Q.38 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?
Q.39 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.
Q.40 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?
Q.41 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 the following figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take π =22/7)
[Hint: Length of successive semicircles is l1, l2, l3, l4, . . . with centres at A, B, A, B, . . ., respectively.]
Q.42 200 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 the following figure). In how many rows are the 200 logs placed and how many logs are in the top row?
Q.43 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 the following figure)
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)]
Q.44 Which term of the AP: 121, 117, 113, . . ., is its first negative term?
[Hint: Find n for an < 0]
Q.45 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.
Q.47 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.
[Hint: Sx–1 = S49 – Sx]
FAQs (Frequently Asked Questions)
1. What is the best way to find the Sum of an arithmetic progression?
To find the Sum of an arithmetic progression, you’ll need to know the first term’s value, the number of terms, and the common difference between each term. To arrive at the final solution, use the formula given below.
S = (n / 2)*(2a+(n−1)d)
How many types of progressions are there in Mathematics?
Ans: In Mathematics, there are three types of progressions. They are: s:
- Arithmetic progression (A.P.)
- Geometric progression (G.P.)
- Harmonic progression (H.P.)