# Sequences and Series

Topic Overview

Most people have met sequences at some stage in their lives. Questions of the form “what is the next number in the list 1, 4, 9, 16, …” frequently appear in quizzes. The list of numbers, separated by commas, is a sequence. In practice, the numbers in the list usually follow some pattern. In the example above, the next number is 25, because it is the square of 5. So what is a series? A series is just what we get when we add some of the terms of a sequence, e.g., 1+4+9+16.

Like most ideas treated mathematically, we find that we have to introduce terminology and notation to allow us to develop fully the properties of series. Also, there are many different kinds of sequence and series. However, we only have to investigate two kinds on our course, namely arithmetic and geometric sequences and series.

Sequences and series are examined in Question 5 on Paper 1 each year. Like most other questions on the first paper, a certain proficiency with algebra is required to answer many of the harder questions on sequences and series. But the algebra content is usually nowhere near that required in algebra itself, functions and differentiation.

Sequence and Series Notation
A sequence is an ordered list of numbers and the sum of the terms of a sequence is a series. The term number is written as a subscript under the T.  The fourth term would be designated T4. The nth term would be designated Tn. The first number in a pattern of numbers is often designated “a” in formulae and could also be designated T1. The letter “d” is used for a constant number that is added to each number in a pattern to get the next one. It is the common difference between adjacent numbers. The letter “r” is used to represent the common ratio or number by which each term is multiplied to get the next one. Sn is represents the sum of the first n numbers in a pattern.

Arithmetic Sequences and Series (also called linear)
A pattern of numbers obtained by adding a common difference to each entry to get the next one is known as an Arithmetic or Linear Sequences or Series. Knowing a and d we can find the value of any term or the sum of the first n terms using the formulae below.

Geometric Sequences and Series
A pattern of numbers obtained by multiplying any term by a common ratio to get the next number in the pattern is known as a Geometic Sequences or Series. Knowing a and r we can find the value of any term or the sum of the first n terms using the formulae below. All these formulas are found on page 22 of the “Formulae and Tables” booklet which will be provided during the Inter Cert and Leaving Cert exams. To know which ones to use, look for the keyword Arithmetic or Geometric in the question.