text: Instasolv
  
  IIT-JEE NEET CBSE NCERT Q&A
  
  4.5/5
  
  # Higher Algebra Hall & Knight Binomial Theorem (Chapter 14) Solutions
  
  Hall and Knight Higher Algebra Solutions for Chapter 14 ‘Binomial Theorem, Any Index’ are prepared to help you in easily solving the exercise questions based on this theorem. These solutions are a perfect guide for Class 12 board exams along with competitive exam preparation like JEE and NEET. The topics covered in this chapter are Euler’s proof of the binomial theorem for a given index. You will learn the general expansion of (1+x)n and the expression (x+y)n, general expansion, and particular cases of (1+x)n and (1-x)-n respectively, and approximation obtained by the binomial theorem. Other crucial topics covered in this chapter include the evaluation of the greatest numerical term in the expansion of (1+x)n, number of homogeneous products of ‘r’ dimensions, terms in the expansion of a multinomial, number of combinations of ‘n’ things taken ‘r’ at a time.
  
  Higher Algebra by Hall and Knight Chapter 14 Binomial Theorem, Any Index includes 85 questions compiled together in 3 exercises. The questions range from the beginner to the advanced level. Practising these questions will also help you understand the patterns of questions that may be asked in the competitive exams such as NEET, JEE Mains or JEE Advanced for this topic of the binomial theorem. You will learn to manage your time effectively by practising these exercise questions of Hall and Knight Higher Algebra Chapter 14 ‘Binomial Theorem’.
  
  The expert Maths teachers at the Instasolv platform have covered Chapter 14 of the Hall and Knight Higher Algebra book thoroughly. The answers to the exercise questions are written in a step by step fashion with an in-depth analysis of each topic covered in the questions. Solving the exercise questions using Instasolv as your source of reference will help you develop the right approach in understanding the concepts of higher algebra.
  
  ## Important Topics Covered under Higher Algebra by Hall and Knight Chapter 14: Binomial Theorem, Any Index
  
  Introduction
  
  In this chapter, you will get to learn the formulae when the values of the index is a negative value or a fractional value. The principle of permanence of the equivalent forms is followed here, which means that in any algebraical product, the order of the product will remain the same even if the quantities involved are whole numbers, fractions, positive, or negative integers.
  
  Binomial Theorem when the Index is a Positive Fraction
  
  We assume the functions and
  
  It can be proved using the permanence of equivalent forms principle that
  
  f(m) x f(n) x f(p)… to k factors=f(m+n+p+… upto k terms)
  
  Therefore, proving that the binomial theorem holds true for positive fractional index.
  
  Binomial Theorem when an Index is a Negative Number
  
  We already know that f(m)f(n)=f(m+n). Now, if we replace m by -n where n is a positive quantity, we get f(-n)f(n)=f(-n+n)=f(0)=1
  
  Therefore, (1-x)-n=1+(-n)x+(-n)(-n-1)x2/1.2+… proving that the binomial theorem holds true for any given negative index.
  
  The convergent and the divergent series are also discussed in this chapter along with their true arithmetic equivalent.
  
  Formula to Evaluate the General Term
  
  Note that the term nCr is no longer used in the general formula, instead, the formula is written in full where n represents a negative or a fractional number. To find the general term, we use the following formula,
  
  Simplest Term in the Expansion of (1-x)-n
  
  When we expand (1-x)-n, all the terms are positive. The simplest term is given by the following expression
  
  ### Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 14: Binomial Theorem, Any Index
  
  1. In exercise 14.a, you will be needed to expand given binomial expressions with fractional or negative values as index up to 4 terms. You will also get to find general terms in various equations given in this exercise.
  2. There are 37 questions in 14.b in which you will be required to find the simplest or the (r+1)th term. You will get many questions based on the applications of the binomial theorem such as some particular cases of (1-x)nand (1-x)-n.
  3. In 14.c, the topics that are covered are the evaluation of the number of homogeneous products of ‘r’ dimensions, terms in the expansion of a multinomial, number of combinations of ‘n’ things taken ‘r’ at a time.
  
  ### Why use Higher Algebra by Hall and Knight Chapter 14: Binomial Theorem, Any Index by Instasolv?
  
  • The Instasolv platform has been made to revolutionise the self-study patterns in students.
  • You can get your doubts related to Binomial Theorem addressed as soon as you stumble upon any of them.
  • The experienced maths faculties on our platform have created the Hall & Knight Algebra Mathematics book with solutions in a step by step manner.
  • These answers are precise and on-point, despite detailed reasoning to each question.
  • Referring to our platform consistently for your doubts will amplify your marks in school exams as well as in the competitive entrance exams, significantly.
  More Chapters from
explanation: The provided text is a detailed description of a chapter from the book "Higher Algebra" by Hall and Knight, specifically focusing on the Binomial Theorem for any index. The text outlines the key topics covered in the chapter, such as the binomial theorem for positive fractional and negative indices, and the evaluation of general terms and homogeneous products. It also mentions the exercises and the types of questions included, which range from beginner to advanced levels. However, the text does not contain a specific math question or a question and answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Instasolv
  
  IIT-JEE NEET CBSE NCERT Q&A
  
  4.5/5
  
  # Higher Algebra Hall & Knight Binomial Theorem (Chapter 14) Solutions
  
  Hall and Knight Higher Algebra Solutions for Chapter 14 ‘Binomial Theorem, Any Index’ are prepared to help you in easily solving the exercise questions based on this theorem. These solutions are a perfect guide for Class 12 board exams along with competitive exam preparation like JEE and NEET. The topics covered in this chapter are Euler’s proof of the binomial theorem for a given index. You will learn the general expansion of (1+x)n and the expression (x+y)n, general expansion, and particular cases of (1+x)n and (1-x)-n respectively, and approximation obtained by the binomial theorem. Other crucial topics covered in this chapter include the evaluation of the greatest numerical term in the expansion of (1+x)n, number of homogeneous products of ‘r’ dimensions, terms in the expansion of a multinomial, number of combinations of ‘n’ things taken ‘r’ at a time.
  
  Higher Algebra by Hall and Knight Chapter 14 Binomial Theorem, Any Index includes 85 questions compiled together in 3 exercises. The questions range from the beginner to the advanced level. Practising these questions will also help you understand the patterns of questions that may be asked in the competitive exams such as NEET, JEE Mains or JEE Advanced for this topic of the binomial theorem. You will learn to manage your time effectively by practising these exercise questions of Hall and Knight Higher Algebra Chapter 14 ‘Binomial Theorem’.
  
  The expert Maths teachers at the Instasolv platform have covered Chapter 14 of the Hall and Knight Higher Algebra book thoroughly. The answers to the exercise questions are written in a step by step fashion with an in-depth analysis of each topic covered in the questions. Solving the exercise questions using Instasolv as your source of reference will help you develop the right approach in understanding the concepts of higher algebra.
  
  ## Important Topics Covered under Higher Algebra by Hall and Knight Chapter 14: Binomial Theorem, Any Index
  
  Introduction
  
  In this chapter, you will get to learn the formulae when the values of the index is a negative value or a fractional value. The principle of permanence of the equivalent forms is followed here, which means that in any algebraical product, the order of the product will remain the same even if the quantities involved are whole numbers, fractions, positive, or negative integers.
  
  Binomial Theorem when the Index is a Positive Fraction
  
  We assume the functions and
  
  It can be proved using the permanence of equivalent forms principle that
  
  f(m) x f(n) x f(p)… to k factors=f(m+n+p+… upto k terms)
  
  Therefore, proving that the binomial theorem holds true for positive fractional index.
  
  Binomial Theorem when an Index is a Negative Number
  
  We already know that f(m)f(n)=f(m+n). Now, if we replace m by -n where n is a positive quantity, we get f(-n)f(n)=f(-n+n)=f(0)=1
  
  Therefore, (1-x)-n=1+(-n)x+(-n)(-n-1)x2/1.2+… proving that the binomial theorem holds true for any given negative index.
  
  The convergent and the divergent series are also discussed in this chapter along with their true arithmetic equivalent.
  
  Formula to Evaluate the General Term
  
  Note that the term nCr is no longer used in the general formula, instead, the formula is written in full where n represents a negative or a fractional number. To find the general term, we use the following formula,
  
  Simplest Term in the Expansion of (1-x)-n
  
  When we expand (1-x)-n, all the terms are positive. The simplest term is given by the following expression
  
  ### Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 14: Binomial Theorem, Any Index
  
  1. In exercise 14.a, you will be needed to expand given binomial expressions with fractional or negative values as index up to 4 terms. You will also get to find general terms in various equations given in this exercise.
  2. There are 37 questions in 14.b in which you will be required to find the simplest or the (r+1)th term. You will get many questions based on the applications of the binomial theorem such as some particular cases of (1-x)nand (1-x)-n.
  3. In 14.c, the topics that are covered are the evaluation of the number of homogeneous products of ‘r’ dimensions, terms in the expansion of a multinomial, number of combinations of ‘n’ things taken ‘r’ at a time.
  
  ### Why use Higher Algebra by Hall and Knight Chapter 14: Binomial Theorem, Any Index by Instasolv?
  
  • The Instasolv platform has been made to revolutionise the self-study patterns in students.
  • You can get your doubts related to Binomial Theorem addressed as soon as you stumble upon any of them.
  • The experienced maths faculties on our platform have created the Hall & Knight Algebra Mathematics book with solutions in a step by step manner.
  • These answers are precise and on-point, despite detailed reasoning to each question.
  • Referring to our platform consistently for your doubts will amplify your marks in school exams as well as in the competitive entrance exams, significantly.
  More Chapters from
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a detailed description of a chapter from the book "Higher Algebra" by Hall and Knight, specifically focusing on the Binomial Theorem for any index. The text outlines the key topics covered in the chapter, such as the binomial theorem for positive fractional and negative indices, and the evaluation of general terms and homogeneous products. It also mentions the exercises and the types of questions included, which range from beginner to advanced levels. However, the text does not contain a specific math question or a question and answer.
  
  Label: no-question