Chapter 4: Sequences and series

The chapter problem: will the village hall roof be paid for?

The village hall needs a new roof. It will cost £9,000.

Two fundraising streams run at the same time. They both start in month 1.

• The quiz night. It raises £150 in month 1. Each month it raises £50 more than the month before.
• The online appeal. It raises £600 in month 1. Each month it raises 80% of the month before, because the appeal fades.

A former resident gives £2,000 at the start. The committee puts the gift in an account that pays 6% compound interest each year.

By the end of the chapter you will answer all of this exactly. You will find how much each stream raises in a chosen month, and how much each has raised after one year. You will find the most the online appeal can ever raise. You will decide whether the fund reaches £9,000, and when. You will also work out what the £2,000 gift becomes, and estimate its value between whole years.

First contact — no working needed yet. Work out the income for months 1, 2 and 3 of each stream. For the quiz night, add £50 each time. For the online appeal, work out 80% of the month before. Write the first three numbers of each stream, then write one sentence saying which stream you expect to raise more over a year.

By the end of this chapter you will be able to

• use factorials and the notation $\binom{n}{r}$;
• expand $(a + bx)^n$ when $n$ is a positive whole number;
• find one term or one coefficient without expanding a whole bracket;
• expand $(1 + x)^n$ when $n$ is a fraction or negative, and state when the expansion is valid;
• use the first terms of an expansion to approximate a value;
• generate a sequence from a formula, or from a rule that links each term to the one before;
• decide whether a sequence increases, decreases or repeats;
• read and write a sum in sigma notation;
• use the standard results for $\sum r$, $\sum r^2$ and $\sum r^3$;
• find the $n$th term and the sum of an arithmetic sequence;
• find the $n$th term and the sum of a geometric sequence;
• find the sum to infinity of a geometric series when $|r| < 1$;
• build a sequence model, state its assumptions, and refine it.

Where it is used

Sequences describe anything that changes in steps. Savings, populations, medicine in the blood and the value of a machine all follow a rule from one value to the next. Compound interest is a geometric sequence. The binomial expansion is how a calculator estimates a power or a root. The coefficient $\binom{n}{r}$ returns in statistics, where it counts the ways a result can happen. Later chapters use these series as the building blocks of calculus.

Your turn to connect it

Think of something in your own life that changes by a rule. It could be a saving plan, a subscription price, a rumour, a phone battery, or a ball that bounces. Write down the starting value, the rule for the next value, and one question you would like the mathematics to answer.

Roadmap

1. 4.1 Binomial expansion for positive integer n; n! and nCr
2. 4.2 Binomial expansion for rational n and approximation
3. 4.3 Sequences from formulae and recurrences
4. 4.4 Sigma notation for sums
5. 4.5 Arithmetic sequences and series
6. 4.6 Geometric sequences and series, and the sum to infinity
7. 4.7 Sequences and series in modelling
8. Review