NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progression Exercise 5.2

Mathematics is an essential topic for students to learn in school. Strong mathematical skills can benefit children in a variety of ways throughout their lives. Besides that, students need Mathematics to understand the principles taught in other subjects, including Physics, Chemistry, etc. A great foundation in Mathematics can therefore help students succeed in other subjects as well. The ideas studied in Class 10 are continued in many higher education courses as well. In higher-level courses of study, a lot of the concepts that are introduced to students in Class 10 are utilised. The concepts learned in Class 10 are crucial for students who want to study Mathematics further in Class 11 and Class 12. Therefore, in addition to helping them perform well on exams, students would get several advantages from having a firm understanding of Mathematics in Class 10. The topics taught in Class 10 are enjoyed by many students. However, reading alone will not be enough to help someone do well on their exams. Practice is the core of Mathematics. As a result, obtaining high marks in this subject requires a lot of effort. According to experts, students should complete the NCERT exercises in order to thoroughly comprehend the chapters. The questions in the exercises, however, are not straightforward. Students frequently struggle to understand some questions because they are so challenging to respond to. They therefore look for  solutions to the exercises’ problems. However, it can be challenging to find NCERT solutions, particularly for students in Class 10. Furthermore, even if it is available, it must be written in a way that is simple for students to understand. Extramarks therefore provides NCERT solutions like NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2.

Class 10 is a pivotal point in a student’s academic career. It prepares students for the more challenging subjects they will study in higher education. Furthermore, marks earned in Class 10 may be used to determine admission to institutes of higher secondary education.Students in Class 10 must do well on the CBSE board exam to have better future opportunities. Students are required to study a variety of challenging subjects in Class 10. They experience a great deal of stress as a result of their preparation, which may negatively impact how well they perform on the board exam. As a result, in order to put forththeir best effort, students must have a sound preparation plan. Moreover, they should have enough study materials and resources for preparation. For this reason, Extramarks offers a variety of study tools that can be used to read over and revise each subject. One of the most important subjects for Class 10 students is Mathematics. There is a lot of work involved in getting good grades in Mathematics. Students must extensively understand each topic. In order to fully understand the topics, they must also practise answering a lot of questions. However, Class 10 students occasionally find it tough to study so many topics. There are various rules and formulas that make students feel overloaded. For this reason, it is advised that students practise NCERT exercises and use the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 and other exercises made accessible on the Extramarks platform.

The internet has a wealth of study resources for students studying Mathematics. Class 10 students may not necessarily require and benefit from each one of these materials to adequately prepare for their exams. Since the CBSE mandates that all of its students use NCERT textbooks, it is imperative that students read the NCERT book. To aid in the development of policies to improve school education, the Union Government formed the NCERT (National Council of Educational Research and Training). It was formed in 1961 and offers guidance on educational policy to the central and state governments. The NCERT conducts extensive research to improve school education in India. One of NCERT’s primary responsibilities is to publish textbooks for students. It also publishes newsletters, journals, and other supplementary materials to ensure that students from all classes receive a quality education. Students can access a variety of digital resources on NCERT’s official website, including the NCERT textbooks in PDF format. The reading of NCERT textbooks is the most crucial part of CBSE students’ test preparation. Furthermore, the NCERT textbooks serve as the foundation for several competitive tests’ syllabi. Experts advise students to read the NCERT textbooks carefully as a result. It is crucial to do the exercises in the book after reading. Students must consequently have access to the NCERT solutions, such as the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2.

Finding the best resource to meet their academic goals is a common challenge for students. The appropriate and essential solution to their educational demands is provided by NCERT solutions. The availability of NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 meets the needs of students who are preparing for their board exams. The fact that the CBSE itself suggests using the NCERT textbook is an additional benefit. Textbooks from NCERT are covered in the CBSE curriculum. The textbooks are a great source of exam-relevant practise questions. Students are frequently baffled by these questions, which makes them less confident in the chapter and the subject. It is essential to comprehend the ideas presented in the first topic. Students who have access to the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 can benefit from having a firm grasp of both fundamental and complicated ideas when attempting the questions in the NCERT textbooks. By finishing it, students can get a strong conceptual understanding of the topic. Moreover, they can assess themselves and make adjustments by comparing their responses with the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2.

All the problems in Maths Class 10 Chapter 5 Exercise 5.2 are answered completely and are without errors. By steadily practising NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2, students may respond to such questions quickly and accurately in exams. By using NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2, students can more accurately apply the concepts given in the chapter. Highly skilled professionals appropriately developed the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2. These solutions are created while keeping in mind the depth of understanding that students in Class 10 possess. Since examiners generally emphasise writing step-by-step answers in exams, the questions in this exercise have been broken into smaller segments to make it easier for students to write answers. Students can achieve the highest scores by using the step-by-step solutions in the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2. Many students may find it difficult to comprehend and memorise the numerous formulas provided in the NCERT textbook, which inhibits them from employing the formulae to answer the exercises’ questions. However, it is easier for students to retain and apply the formulas when they practise the stepwise solutions in the NCERT Solutions for Class 10 Maths Chapter 5 Exercise 5.2.

NCERT Class 10 Maths Solutions for Chapter 5 Arithmetic Progression Ex 5.2

One of the most common reasons why students struggle to complete the question paper within the allocated time is a lack of practice. There are also several questions from the NCERT book on the exam paper. With the help of NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2, students can learn how to structure their answers and complete the question paper even before the exam’s deadline. The NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 are thus available at Extramarks. Students may obtain the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 whenever they want because they are made available in PDF format. One can quickly download the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 from the Extramarks website and smartphone application.

Access NCERT Solutions for Class 10 Maths Chapter 5 – Arithmetic Progression

Students need to have access to all the required study materials in order to do well on the board exams. Access to the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 might not always be possible as students sometimes face connectivity issues with the internet. Even if it might not always be possible, having access to the internet is crucial. For students who want to continue learning, the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 in PDF format must be available. Extramarks offers a comprehensive solution for a student’s educational needs. Even though NCERT textbooks make excellent study resources, students still have trouble finding trustworthy NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2. The detailed and accurate NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 are provided by Extramarks. Additionally, it provides students with access to a comprehensive learning programme that combines the benefits of virtual classroom instruction with study aids like NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2.

NCERT Solutions Class 10 Maths Chapter 5 Exercise 5.2

One of India’s main national-level educational boards is the Central Board of Secondary Education (CBSE). The CBSE has thousands of schools affiliated with it. It is the largest education board in India in terms of overall enrolment. Millions of students are registered with the CBSE. The board exams are taken in Class 10. The board exam in Class 10 is a difficult process for students as they have no prior experience with it. Many students perform poorly because they do not study sufficiently for the exams. They struggle to achievetheir desired goals. They also struggle to produce the desired results. The best materials are offered on Extramarks to assist students in making efficient preparations andand overcoming exam anxiety. It provides a range of study tools that could help students get better prepared for exams. With Extramarks, students can get a complete study solution.

The most crucial resource for reviewing questions for the CBSE board exams is NCERT exercises. Therefore, it is usually recommended for Class 10 students to solve the questions from the NCERT textbook. They also require the answers to the questions, so they can examine their answers. Therefore, the NCERT solutions give students the assistance they require while getting ready for the board exams. Using the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 will help students apply the concepts from the chapter in a more easy and accurate manner. Highly skilled teachers have produced detailed and error-free NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2. They develop these answers while taking into account the comprehension skills of students in Class 10. The problems in this exercise have been divided into smaller, more manageable parts to make writing answers easier. Examiners frequently emphasise the requirement for detailed, stepwise responses. Students should follow the instructions in the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 in order to get ideal scores on exams.

Students who are preparing for board exams must work hard and intelligently. To get the most out of their preparation, students need to create a sound approach. They need to create a study timetable stating the amount of time needed for each subject and make an effort to adhere to it at all times. They must arrange every aspect of their preparation in advance. Students should download the NCERT Solutions for Class 10 Maths, Chapter 5, Exercise 5.2 ahead of time in order to use them all year, even if they do not have an internet connection.First, they should carefully study each topic in the course syllabus. They should study the textbook’s questions and make notes on crucial topics. Likewise, they must try to complete each question in the exercises on their own. When they cannot, they should use the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 available on Extramarks. Additionally, they should pay close attention to the formulas involved in the questions. They can record the methods they followed to respond to various kinds of questions in their notes. Following that, the topics should be revised, with a greater emphasis placed on their weak points. Finally, in order to fully understand the material, they should practise taking mock exams.

Important Topics under NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progression Exercise 5.2

The NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 cover the topic of the nth Term of an AP. This topic is one of the most important  from an examination perspective. Studying this topic will help students find the nth term of an arithmetic progression, which is also called the general term of the arithmetic progression.

Importance of Arithmetic Progressions

Arithmetic Progressions are one of the most important concepts in Mathematics. Having knowledge of arithmetic progressions can help students solve various problems in daily life. Studying this chapter can help students find the succeeding terms in a sequence of numbers that follow a specific pattern.

NCERT Solutions for Class 10 Maths Exercise 5.2 – Arithmetic Progression

Before appearing for the board exams, students need access to a range of questions as well as clearly explained solutions. The majority of question banks base their questions on NCERT textbook problems.. When students have access to the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2, answering questions is easier. They capture the concepts more quickly because of this. Quick learners and graspers of new concepts feel more prepared for any exam. Additionally, this gives students more time to study while retaining mental concentration on other aspects of their schooling. By understanding the concepts outlined in the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2, students can advance in their preparation. Having access to the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 that are tailored to students’ needs can be a useful study tool, despite the chapter’s occasionally challenging and complex content. Utilising these study tools can make studying Arithmetic Progression a simpler subject. Students can practise by completing the NCERT questions if they want to progress consistently while finishing other papers or problems. Study tools like NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 by Extramarks concentrate on a specific chapter. There are several learning modules and a wide range of study resources available to students who desire to perform better on the board exam, including chapter-by-chapter worksheets and sample tests. Therefore, Extramarks serves as a one-stop destination for testing, practice, and study. The fact that Extramarks’ NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 are created and curated by skilled subject professionals guarantees a flawless learning experience for all students.

All students are required to read the NCERT books in their entirety. Students must follow the CBSE’s published syllabus while they are studying. They should also organise their answers in accordance with the CBSE marking criteria if they want to get the highest possible score. The NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 were created by teachers with considerable experience in the CBSE board exams. They must follow the most recent CBSE guidelines when creating the NCERT Solutions for Class 10 Maths, Chapter 5 Exercise 5.2.According to the most recent guidelines, the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 are frequently revised and updated. The most recent curricular changes implemented by the CBSE are not causing concern for students who follow the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2. On the Extramarks portal, learning resources are arranged by chapters and subjects for every subject, including Mathematics. The experts in the field put a lot of work into clarifying and simplifying concepts for exam preparation.

One of the most crucial activities for students preparing for the Class 10 board exams is to complete the NCERT textbook exercises in the chapter on Arithmetic Progression. For optimum effectiveness, students should apply a few streamlined strategies when attempting to solve the problems in this lesson.Exams in Mathematics require students to prepare detailed answers. They should therefore be aware of the procedures to follow while writing answers in order to receive the highest marks for the questions. First, students should attentively study and comprehend the chapter’s explanations in the textbook. They can use Extramarks’ study materials if they are having trouble comprehending the subject matter. Once they are familiar with the fundamentals, they should work through the NCERT examples and comprehend the methods involved in answering the questions. They will gain knowledge of how to deal with the problems. Then they should try to respond to the questions in the exercises in order. Before reading the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2, they should try to find the answers to the questions on their own. They should try to remember all the possible approaches to the problems. They should verify the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 after answering each question to make sure their answers are accurate. Furthermore, they should use the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 to correct their wrong answers. The NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 should be accurately followed by the students. They must try to solve these problems repeatedly until they can do so successfully. To find the answers to the problems they are unable to answer, students could also check the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2. They should redo the exercises in the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 once they have acquired the solutions. They will be able to understand the questions better and, as a result, respond to them more quickly and effectively. Additionally, they need to make an effort to find solutions as soon as possible. By answering these kinds of questions in the board examinations more quickly and accurately, they would be able to finish the exam paper on time.

5.2 Arithmetic Progression

Arithmetic Progression is one of the most crucial topics that Class 10 students need to learn. There are various situations in daily life where knowledge of arithmetic progressions can be applied. There are situations where people have to deal with numbers that follow a certain pattern. Studying this chapter will help students in dealing with such situations easily. Class 10 Maths Chapter 5 Exercise 5.2 includes concepts like general terms of an AP and common differences. It is crucial for students to remember the formula for the general term of an AP. This will help them solve many questions in the exercises. Students can also refer to the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 for additional help.

nth Term of an AP

Class 10th Maths Exercise 5.2 deals with the topic nth term of an AP. Before solving the questions in this exercise, students need to thoroughly understand the concepts in the textbook to understand the significance of the nth term of an Arithmetic Progression. Students also need to study the topics before this one. They should first understand the basic concepts of arithmetic progression to understand this topic. A thorough understanding of the topic will help students determine the general term of an arithmetic progression. The exercise also includes numerous application-based questions. Students should practise more such questions, as they are often found in the board examinations. Students can refer to the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 to learn all the steps and methods of solving these questions.

Sum of the First N Terms of an AP

The last topic of Chapter 5 is based on the Sum of the First n Terms of an AP. Studying this topic can be very helpful for students in their day-to-day lives. By studying this topic, students will be able to find the sum of the first n terms of an AP. Students need to remember the formula that is used to find the sum, as it is often applied while solving questions in the exercises.

Summary

The chapter on Arithmetic Progression is one of the most important chapters for students preparing for the board examinations. By studying this chapter, students can learn various concepts related to Arithmetic Progression. Students can understand what an arithmetic progression is. They can understand the concept of common differences in an arithmetic progression. Students can understand when a sequence can be called an arithmetic progression. Students can also learn to find the general term and the sum of the first n terms of an arithmetic progression. As various types of questions, including application-based questions, are asked in this chapter, students need to practise the exercises in the chapter to be able to solve any question that may appear in the board examinations.

Key Features of NCERT Solutions for Class 10 Maths Exercise 5.2

For students in Class 10, the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 are regarded as a crucial study aid as they get ready for the board exams. For students in Class 10, the CBSE recommends the NCERT Mathematics textbook. Additionally, the NCERT textbook is advised in a number of other boards’ syllabi. Therefore, the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 from Extramarks will be helpful to students from different boards as well as those from the CBSE. Along with the board exams for Class 10, competitive tests like Olympiad also include Arithmetic Progressions. The outcomes of these exams also have a significant impact on a student’s career. While preparing for multiple exams, students have so much stress that it wears them out and occasionally causes them to lose motivation. However, studying with the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 provided by Extramarks can assist students in understanding the concepts completely and remembering them for a long time. They consequently have more confidence while trying to study for several exams. Less effort is required to succeed in the competitive exams for those who thoroughly study the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 and other exercises. The NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 thoroughly explain the solutions to assist students in passing such challenging tests. They can improve their fundamentals by using the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2. As a result, whether preparing for the board exam or some competitive examination, the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 are very crucial.

Along with other study materials on Extramarks, the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 cover all the crucial topics listed on the CBSE syllabus for board exams. The NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 are written by experienced educators to help students get ready for problems that will arise in the CBSE board exams. Students can prepare for competitive exams like Olympiad by practising the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2. To fully understand the problems, students must practise the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 numerous times. Students must regularly practise the NCERT textbook exercises in order to complete the question paper within the allotted time. One of the most common reasons why students fail to finish the question paper in the allotted time is a lack of practice. The exam paper also includes a number of questions from the NCERT book. Students learn how to prepare the answers by using the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2, which helps them complete the question paper before the exam s time limit.

NCERT Solutions for Class 10 Maths Chapter 5 Exercises

One of the few comprehensive and authentic NCERT solutions on the internet is  Extramarks’ NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2. The NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 are available on many different websites, but some students and teachers are unsure of how reliable they are. Students should take advantage of reliable study materials because board exams are very crucial. Simple language is used to describe the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 and other exercises. Students can learn the proper format for structuring their answers. The NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2 offer the most effective and uncomplicated answers to the chapter’s mathematical problems. In addition to the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2, Extramarks also provides a number of learning resources for Class 10 Mathematics. The study resources make it simpler to comprehend the ideas required to finish the exercises. Revision of the chapter is also facilitated by these learning resources. They are written by qualified, experienced academics with years of expertise in a format that is straightforward and understandable. Practice worksheets, solved exemplar problems, revision notes, and other helpful materials are available on Extramarks. As a result, students can save time by finding all they need on one site and can make use of everything they need for their studies.

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Students in Class 10 who are preparing for their board exams can get NCERT Solutions Class 10 for all of their subjects in addition to the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2. In order to meet the needs of both Hindi and English medium students, the NCERT Solutions Class 10 are offered in both languages. NCERT Solutions Class 12 for Mathematics and all other disciplines are also available for Class 12 students preparing for the Class 12 board exams. For Class 11 students, Extramarks offers NCERT Solutions Class 11 Mathematics and other subjects. Students in other classes can also access NCERT Solutions in Hindi and English for all subjects. They can get NCERT Solutions Class 10 on the Extramarks website and mobile app, along with NCERT Solutions Class 8, NCERT Solutions Class 9, NCERT Solutions Class 7, and NCERT Solutions Class 6. Additionally, Extramarks offers NCERT solutions for kids in primary classes, including NCERT Solutions Class 4, NCERT Solutions Class 5, NCERT Solutions Class 3, NCERT Solutions Class 2, and NCERT Solutions Class 1. On the Extramarks website and mobile app, students may download the PDF version of the NCERT Solutions For Class 10 Maths Chapter 5 Exercise 5.2.

Q.1

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, 12, 2, . . ., is (A) 28 (B) 22 (C) 38 (D) 4812

Ans.

(i)The given AP is: 10, 7, 4, . . . Here, a=10, d=710=3, n=30We know that an=a+(n1)d a30=10+(301)(3)=1087=77Hence, the correct answer is (C).(ii)The given AP is: 3, 12, 2, . . .Here, a=3, d=212=52, n=11We know that an=a+(n1)d a11=3+(111)×52=3+25=22Hence, the correct answer is (B).

Q.2

InthefollowingAPs,findthemissingtermsintheboxes:i 2,,26ii ,13,,3iii 5, , ,912iv4,, ,, ,6v ,38, , , ,22

Ans.

iThegivenAPis:2,,26Letthemissingtermist.Then,wehavet2=26t2t=26+2=28t=282=14iiThegivenAPis:,13,,3Letthecommondifferenceisd.Then,wehavea4a2=2d=3132d=10d=102=5Therefore,firstterm=a1=a2d=135=13+5=18thirdterm=a3=a4d=35=3+5=8Hence,thegivenAPis:18,13,8,3iiiThegivenAPis:5,,,912Letthecommondifferenceisd.Then,wehavea2=a1+d,a3=a2+d=a1+d+d=a1+2d,a4=a3+d=a1+2d+d=a1+3d192=5+3d3d=1925=92d=92×13=32Therefore,missingtermsofthegivenAPare:a2=a1+d=5+32=132,a3=a2+d=132+32=162=8Hence,thegivenAPis:5,132,8,912ivThegivenAPis:4,,,,,6Letthecommondifferenceisd.Then,wehavea6=a1+5d6=4+5d5d=6+4=10d=105=2Therefore,missingtermsofthegivenAPare:a2=a1+d=4+2=2a3=a1+2d=4+2×2=0,a4=a1+3d=4+3×2=2,a5=a1+4d=4+4×2=4Hence,thegivenAPis:4,2,0,2,4,6vThegivenAPis:,38,,,,22Letthecommondifferenceisd.Then,wehavea6=a1+5da6=a1+d+4d=a2+4d22=38+4d4d=2238=60d=604=15Therefore,missingtermsofthegivenAPare:a1=a2+d=3815=38+15=53,a3=a1+2d=53+2×15=23,a4=a1+3d=53+3×15=8a5=a1+4d=53+4×15=7Hence,thegivenAPis:53,38,23,8,7,22

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

Ans.

The given AP is: 3, 8, 13, 18, . . . Let nth term of the given AP is 78. We know that an=a+(n1)d78=3+(n1)5(n1)5=783=75n1=755=15n=15+1=16Therefore, 16th term of the given AP is 78.

Q.4

Find the number of terms in each of the following APs:(i) 7, 13, 19, . . . , 205 (ii) 18, 1512, 13, . . . , 47

Ans.

(i) 7, 13, 19, . . . , 205 Let the given AP has n terms. We know that an=a+(n1)d205=7+(n1)6(n1)6=2057=198n1=1986=33n=33+1=34(ii) 18, 1512, 13, . . . , 47Let the given AP has n terms. We know that an=a+(n1)d47=18+(n1)(31218)=18+(n1)(52)(n1)(52)=4718=65n1=65×25=26n=26+1=27

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

Ans.

The given AP is  11, 8, 5, 2, . . .Let nth term of the given AP is 150.We know that an=a+(n1)d150=11+(n1)(811)=11+(n1)(3)(n1)(3)=15011=161n1=161×13=5323n=5323+1=5423But 5423 is not a natural number. Therefore, 150 is nota term of the given AP.

Q.6 Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.

Ans.

We have a11=38 and a16=73Or a11=a1+10d=38                          ...(1) and     a16=a1+15d=73                          ...(2) Subtracting (1) from (2), we get          5d=35        d=7On putting this value of d in (1), we get        a1+10d=38 a1+10×7=38 a1=3870=32Now,a31=a1+30d=32+30×7=32+210=178

Q.7 An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.

Ans.

We have a3=12 and a50=106Or a3=a1+2d=12                           ...(1) and     a50=a1+49d=106                          ...(2) Subtracting (1) from (2), we get          47d=94        d=9447=2On putting this value of d in (1), we get        a1+2×2=12 a1=124=8Now,a29=a1+28d=8+28×2=8+56=64

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

Ans.

We have a3=4 and a9=8Or a3=a1+2d=4                           ...(1) and     a9=a1+8d=8                                ...(2) Subtracting (1) from (2), we get          6d=12        d=126=2On putting this value of d in (1), we get        a1+2×(2)=4 a1=4+4=8Let nth term of the given AP is zero. Then, we have      an=a1+(n1)d0=8+(n1)(2)n1=82=4n=4+1=5Hence, 5th term of the given AP is zero.

Q.9 The 17th term of an AP exceeds its 10th term by 7. Find the common difference.

Ans.

We have a17=a10+7 a17a10=7a1+16d(a1+9d)=7 [an=a+(n1)d]7d=7d=1Therefore, the common difference is 1.

Q.10 Which term of the AP: 3, 15, 27, 39, . . . will be 132 more than its 54th term?

Ans.

Let nth term be 132 more than 54th term of the given AP.We have an=a54+132 ana54=132a1+(n1)d(a1+53d)=132 [an=a+(n1)d](n54)d=132n54=132d=13212 [The given AP is: 3, 15, 27, 39, ...]n=11+54=65Therefore, 65th term is 132 more than 54th term of the given AP.

Q.11 Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?

Ans.

It is given that two APs have the same common difference.Let an and bn represent the terms of the first and secondAP respectively for all natural numbers n.We have a100b100=100 a1+99db199d=100                               [an=a+(n1)d]a1b1=100 Now,a1000b1000=a1+999db1999d=a1b1=100 Therefore, the difference between 1000th terms of the twoAPs is 100.

Q.12 How many three-digit numbers are divisible by 7?

Ans.

The smallest three-digit number divisible by 7 is 105.The largest three-digit number divisible by 7 is 994.Three-digit numbers divisible by 7 are:105, 112, 119, ...,994These multiples of 7 form an AP with common difference 7,first term 105 and last term 994. Thus, we havea1=105, an=994 and d=7.We know that        an=a1+(n1)d994=105+(n1)7(n1)7=994105=889 n1=8897=127n=127+1=128 Therefore, 128 three-digit numbers are divisible by 7.

Q.13 How many multiples of 4 lie between 10 and 250?

Ans.

Multiples of 4 between 10 and 250 are:12, 16, 20, ...,248These multiples of 4 form an AP with common difference 4,first term 12 and last term 248. Thus, we havea1=12, an=248 and d=4.We know that        an=a1+(n1)d248=12+(n1)4(n1)4=24812=236n1=2364=59n=59+1=60Therefore, 60 multiples of 4 lie between 10 and 250.

Q.14 For what value of n, are the nth terms of two APs: 63, 65, 67, . . . and 3, 10, 17, . . . equal?

Ans.

Let nth term of the given APs are equal.Let an and bn represent the terms of the first and secondAP respectively for all natural numbers n. Also, let d1 and d2 arecommon diffrences of first and second AP respectively.The given two APs are:63, 65, 67, . . . and 3, 10, 17, . . .We have an=bn a1+(n1)d1=b1+(n1)d2                          [an=a+(n1)d]63+(n1)2=3+(n1)7(n1)2(n1)7=3635(n1)=60n1=605=12n=12+1=13Therefore, 13th terms of the given APs are equal.

Q.15 Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.

Ans.

We have     a3=16 and a7=a5+12a1+2d=16 and a7a5=12                        [an=a+(n1)d]a1+2d=16 and a1+6da14d=12    a1+2d=16 and 2d=12    a1+2d=16 and d=6    a1+2×6=16 and d=6a1=4 and d=6       Therefore, the required AP is: 4, 10, 16, 22, ...

Q.16 Find the 20th term from the last term of the AP: 3, 8, 13, . . ., 253.

Ans.

Here, a=3, d=83=5, l=253 where l=a+(n1)dTo find the 20th term from the last term, we will find thetotal number of terms in the AP.So, 253=3+(n1)5i.e., 2533=(n1)5i.e., n1=2505=50or n=50+1=51So, there are 51 terms in the given AP. The 20th term from the last term will be the 32nd term(51-20+1) from the beginning.So, a32=3+31×5=3+155=158i.e., the 20th term from the last term is 158.

Q.17 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.

Ans.

Here,    a4+a8=24a+3d+a+7d=242a+10d=24a+5d=12                                             ...(1)Also,      a6+a10=44a+5d+a+9d=442a+14d=44a+7d=22                                            ...(2)We subtract (2) from (1) and get     2d=10d=5On putting this value of d in (1), we get      a+5×5=12a=1225=13First three terms of the AP are a, a+d and a+2d i.e.,13, 13+5 and 13+2×5 i.e., 13, 8 and 3.

Q.18 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?

Ans.

Subba Rao’s annual salary in 1995 was ₹ 5000.He received an increment of ₹ 200 each year.So, Subba Rao’s annual salaries forms an AP witha=5000 and d=200.Let 7000 is nth term of this AP.We know that         an=a+(n-1)d7000=5000+(n-1)200(n-1)200=7000-5000n-1=2000200=10n=10+1=11Therefore, in 11th year, Subba Rao’s salary reach ₹ 7000.

Q.19 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.

Ans.

..So, s forms an AP witha=5 and d=..It is given that in nth week her weekly savings become ..an=.We know that         an=a+(n1)d.=5+(n1)×.(n1)×.=.5n1=15.75.=9n=9+1=10

Q.20

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) 115 , 112 , 110 , . . ., to 11 terms.

Ans.

(i)Here, a=2,  d=a2a1=72=5,  n=10We know that sum of first  n terms of an AP is given by            S=n2[2a+(n1)d]So,    S=102[2×2+(101)5]=5[4+45]or S=245So, the sum of the first 10 terms of the given AP is 245.(ii) Here, a=37,  d=a2a1=33(37)=4,  n=12We know that sum of first  n terms of an AP is given by            S=n2[2a+(n1)d]So,    S=122[2×(37)+(121)4]=6[74+44]or S=180So, the sum of the first 12 terms of the given AP is 180.(iii)Here, a=0.6,  d=a2a1=1.70.6=1.1,  n=100We know that sum of first  n terms of an AP is given by            S=n2[2a+(n1)d]So,    S=1002[2×0.6+(1001)×1.1]=50[1.2+108.9]or S=5505So, the sum of the first 100 terms of the given AP is 5505.(iv) Here, a= 115,  d=a2a1=112115=160,  n=11We know that sum of first  n terms of an AP is given by            S=n2[2a+(n1)d]So,    S=112[2×115+(111)×160]=112[215+16]or S=3320So, the sum of the first 11 terms of the given AP is 3320.

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