CHAPTER 12 - Topics in Algebra

Abstract

This chapter focuses on several topics in algebra. An arithmetic sequence—also called an arithmetic progression—is a sequence such that each term differs from its predecessor by a fixed number called the common difference. Symbolically, the nth term in an arithmetic sequence can be written by a recursive formula an = an − 1 + d and (2) a formula an = a1 + (n − 1) d, which relates an to a1. Adding the terms of a an arithmetic sequence results in a series. A geometric sequence or progression is a sequence such that each term is obtained by multiplying the previous term by a fixed number. In symbols, an = r • an − 1 for n > 1, where r is the common ratio. Geometric means are the terms between the first and last terms. The principle of mathematical induction is a method of proving certain statements involving the natural numbers. The binomial expansion of (a + b)n where n is any natural number is simply the result of carrying out the multiplication. Pascal's triangle is one way of determining the coefficients of the terms of the expansion. From the binomial theorem, the sum of the exponents in any term of the binomial expansion equals n and the product of the first n natural numbers is n factorial, which in symbols is written as n!. From the concepts of permutations and combinations, if one event can occur in m ways and a second event can happen in n ways, then both events can happen in m • n ways.