# Sequences and Series

### Year 11 Mathematics

## Syllabus References

## Approximate schedule of lessons

## 1. Sequences

A sequence is a set of numbers in a specific order. In years 7 to 10, you will have met these in topics dealing with number patterns. A number pattern is a sequence. Some examples of sequences are:

a) 2, 4, 6, 8, 10, 12, …

b) 1, 1, 2, 3, 5, 8, 13, ….

c) 0, 1, 4, 9, 16, 25.

d) 3, -6, 12, -24, 48.

A sequence may be finite or infinite. In examples (a) and (b), the sequences go on forever, and so are called infinite sequences. In examples (c) and (d), the sequences terminate at 25 and 48 respectively, and so are called finite sequences.

Each number in the sequence is called a **term** of the sequence. In example (a), the first term is 2, the second term is 4, the third term is 6, and so on. The notation for this is shown below.

Tn is the nth term and is called a general term in the sequence

- Complete Exercise 1

## 2. Revision of algebraic skills

- Read page 2 of booklet
- Complete revision exercises 2.1, 2.4, 3.1, 3.2

## 3. Arithmetic Sequence

Arithmetic Sequence

In an arithmetic sequence, the terms increase (or decrease) by the same amount. Each term is a constant amount more (or less) than the previous term. Some examples are:

1. 5, 7, 9, 11, 13, ...

2. -13, -6, 1, 8, 15, …

3. 7, 4, 1, -2, -5, …

4. 20, 19½ , 19, 18½ , 18, …

The difference between consecutive terms is the same, and so is called the **common difference**. In the above examples, the common differences are: 2, 7, -3, -½.

Notice that when the sequence decreases, the common difference is negative.

- Read p. 3 of booklet
- Complete ex. 3

## 4. The terms of an Arithmetic Sequence

- Make notes on and read p. 3-4 of booklet
- Complete Ex 4 on p. 5

## 5. Solving Linear Inequations

- Work through the rice distribution problem on p. 5
- Read and make notes on p. 6
- Complete Ex 5

## 6. Simultaneous Linear Equations

- Work through the water tank problem on p. 6
- Work through examples on p. 7
- Complete Ex 6 on p. 8

## 7. Geometric Sequences

- Work through the microbe problem on p. 8
- Work through example 12 and 13 on p. 8-9
- Complete Ex 7

## 8. Terms of a Geometric Sequence

- Read through p. 9-10 and make notes
- Complete Ex 8

## 9. Solving Exponential Equations by Trial and error

- Read through The Betting Game problem on p. 11
- Work through example 15 & 16
- Do Ex 3.6 of textbook
- Complete Ex 9 on p. 12
- Work through example 15
- Complete Ex 10 on p. 13

## 10. Solving simultaneous non-linear equations

In general, we can use either the substitution method or the elimination method. For the type of equations above, the elimination method is best.

- Work through example 18 and 19
- Complete Ex 11 on p. 14

## 12. Series

Series

A series is the **sum of terms** of a sequence.

This is a sequence: 8, 4, 2, 1, ...

THis is the corresponding series: 8 + 4 + 2 + 1 + ...

The symbol Sn is used the represent the sum of "n" terms

Sigma Notation

Mathematicians often use shorthand notation to simplify the writing of an expression. Sigma notation can be used to simplify the writing of series. Sigma (S) is the Greek letter corresponding to our letter, S, and is used to represent the ** s**um of numbers.

- Make notes on p. 14-15
- Complete Ex 12

## 13. Sum of an Arithmetic Series

- Read through the proof on p. 16 and learn the formula on p. 17
- Work through example 22 and 23
- Complete Ex 13 on p. 17

## 14. Quadratic Equations

- Work through the pile of logs problem on p. 18
- Read through notes on quadratic equations
- Complete examples and problems involving the factorisation method on p. 18-19
- Complete examples and problems involving the formula method on p. 19-20
- Complete examples on p. 20-21
- Complete Ex 16

## 15. Sum of a Geometric Series

- Read through p. 22 and learn the formula for the sum to 'n' terms of a geometric series
- Work through example 30
- Complete Ex 17
- Work through the theatre seating problem on p. 23-24
- Complete Ex 18 (Review)