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.

Topic Structure

The study of Senior Cycle Sequences and Series can be divided into the following sections:Sequence and Series Notation
The term number is written under the T
Whatever replaces n on the LHS also replaces n on the RHS
Numbers are separated by + signs instead of commas
Whatever replaces n on the LHS also replaces n on the RHS

Arithmetic Sequences and Series
This condition must hold forever
Knowing a and d we can find anything
You must learn this formula
A is 4 and d is 5
You must learn this formula

Geometric Sequences and Series
This condition must hold forever
Knowing a and r we can find anything
You must learn this formula
A is 4 and r is 3
You must learn this formula

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.

 

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