Chapter 6

Sequences and Series

Chapter description
In this chapter we look at different patterns in sequences of numbers, and how they might be described. We discover how it is possible to find the sum of the terms of some of these sequences, and find some practical applications of these sums.
We begin to see how infinite quantities of things behave through looking at what happens if we have very large numbers of them. Endless quantities of things have to be treated with great caution, so I show you some examples of what can happen otherwise.
The chapter is divided into the following sections:

1. Patterns and Formulas
   1. Finding patterns in sequences of numbers.
   2. How to describe these patterns mathematically.

2. Arithmetic Progressions
   1. What are they?
   2. Finding a rule for summing A.Ps
   3. The arithmetic mean or `average'
   4. Solving a typical problem
   5. A summary of the results for A.Ps

3. Geometric Progressions. (G.Ps.)
   1. What are they?
   2. Summimg geometric progressions
   3. The sum to infinity of a G.P.
   4. What do `convergent' and `divergent' mean?
   5. More examples using G.Ps. Chain letters.
   6. A summary of the results for G.Ps.
   7. Recurring decimals and writing them as fractions.
   8. Compound interest: a faster way of getting rich.
   9. The geometric mean
   10. Comparing arithmetic and geometric means.
   11. Thinking point: what is the fate of the frog down the well?

4. A compact way of writing sums: the sigma notation.
   1. What does the sigma stand for?
   2. Unpacking the sigmas.
   3. Summing by breaking down to simpler series.

5. Partial Fractions
   1. Introducing partial fractions for summing series.
   2. General rules for using partial fractions.
   3. The cover-up rule.
   4. Coping with possible complications.

6. The fate of the frog down the well.

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