# Higher Algebra Hall & Knight Summation of Series (Chapter 29) Solutions

Hall and Knight Higher Algebra Solutions for Chapter 29 ‘Summation of Series’ have been prepared to assist you in solving the questions in the exercises of this chapter without a hitch. This chapter of Higher Algebra by Hall and Knight covers topics such as the product of n factors in an Arithmetic Progression, the reciprocal of the product of n factors in an Arithmetic Progression, method of subtraction, expression of the product u_{n} as the sum of factorials. In this chapter, you will also learn about polygonal and figurate numbers, Pascal’s triangle, method of differences, and further cases of recurring series.

Hall & Knight Algebra Mathematics book with solutions for Summation of Series consists of 85 questions divided into 3 exercises. These questions are based on the application of the topic in contexts of physics and maths. On solving these questions, you will learn how to find the sum of a series up to the n^{th }term, and the product of n factors in an Arithmetic Progression. These questions will enhance your time management skills for competitive entrance exams such as JEE and NEET. You will be able to boost your score exceptionally well if you solve each question in the exercises thoroughly.

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## Important Topics for Hall and Knight Higher Algebra Solutions Chapter 29: Summation of Series

**Introduction**

If we assume two functions of r and r-1, and the difference of these functions given as v_{r}–v_{r-1} yield the r^{th} term, u_{r} of the given series, denoted as u_{1}+u_{2}+u_{3}+…+u_{n}, then the Sum, Sn can be given mathematically as,

S_{n}=(v_{1}–v_{o})+(v_{2}–v_{1})+(v_{3}–v_{2})+…+(v_{n-1}–v_{n-2})+(v_{n}–v_{n-1})

We can also achieve a suitable transformation by using the methods explained in the earlier chapters to separate un into partial fractions.

### Sum of n Terms of a Series in which the Terms are Composed of r Factors

Lets us assume that the first of these given factors for several terms are in the same arithmetic progression and the series be denoted as

u_{1 }+ u_{2 }+ u_{3 }+ …+ u_{n}

Then Sum, where C is the constant, independent of n.

Or, S_{n} = n (2n^{3} + 8n^{2} + 7n-2

### Sum of n Terms of a Series in which the Terms are Composed of Reciprocal of the Product of r Factors

Lets us assume that the first of these given factors for several terms are in the same arithmetic progression and the series be denoted as

**Method of Subtraction**

When the above rules are directly applicable in a given series, then rather than quoting the rule, we can directly term these as ‘Method of Subtraction’

**Polygonal Numbers**

If we assume the expression of the sum of n terms in an arithmetic progression

with the first term as 1 and common difference as b such that

b=0,1,2,3… we will get,

These numbers are termed as the a^{th} terms of the Polygonal numbers of second, third, and so on, orders

**Figurate Numbers**

Consider the series, 1, 1, 1,… If we assume a series with the nth as the sum of this given series, we will get the new series as,

1, 2, 3, 4, 5, …

Now, taking the sum of the n terms of this new series will yield,

1, 3, 6, 10, 15, …

Similarly, if we proceed in this manner, we will obtain successive series such that in any of those, the n^{th} term is the sum of the n terms of the previous series. Such successive series is termed as Firgurate numbers of first, second, and so on, order.

**Pascal’s Triangle Rule**

All the numbers are such that each is the sum of the number immediately above it and the number which is immediate to the left of it.

### Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 29: Summation of Series

- Exercise 29.a consists of 23 questions in which you will be required to find the sum of given series to n terms and to infinity. You will also get to solve word problems related to this topic in the exercise questions.
- The questions in exercise 29.b are based on the method of differences to find the sum of n terms of the given series and the generating functions of the series.
- Exercise 29.c comprises of questions including more complex series and applying more advanced concepts to find the general term and the sum of n terms.

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