Exploring Number Sequences

Create the lesson plan using the following information -1. Sequences

A sequence is a pattern of numbers or shapes that follow a rule. The numbers in a sequence are called terms.

A sequence is like a set, except:

The terms are in order (with sets, the order does not matter). The same value can appear many times (with sets, the same value appears only once). Infinite or finite: When the sequence goes on forever, it is called an infinite sequence. Otherwise, it is a finite sequence. A sequence that is infinite will have three dots placed after the last term listed.

Examples of sequences:

Sequence Description 1, 2, 3, 4 ... The sequence of all natural numbers (an infinite sequence) 10, 15, 20, 25 ... A sequence of multiples of 5 from 10 onwards (an infinite sequence) 1, 3, 5, 7 The sequence of the first 4 odd numbers (a finite sequence) 1, 2, 4, 8, 16, 32 ... The sequence is an infinite sequence where every term doubles. a, b, c, d, e The sequence of the first 5 letters of the alphabet a, p, p, l, e The sequence of letters in the word 'apple' 1, 2, 1, 2, 1, 2, ... An infinite sequence of alternating 1s and 2s 2. Extending Sequences

To extend a given sequence means to use the pattern rule to write the number or pattern that would come next in that particular sequence. To be able to extend a sequence, we first need to know the rule; that is, the first step. Once we have done that, we can work out what the next term is in the sequence.

Worked Example 1

Extend the following sequence by finding the next 3 terms:

11, 15, 19, 23 ... Solutions to Worked Example 1:

The first step is to find the rule of the sequence. We see that each term is 4 bigger than the previous term. The rule is: Add 4 to the previous term to get the next term. So, to get the next term, add 4 to 23 to get 27. The term thereafter will be 27 + 4 = 31, and the one after that, 31 + 4 = 35.

The completed set is:

11, 15, 19, 27, 31, 35 …

Worked Example 2

Write down the next two terms in each sequence.

1. 29, 25, 21, 17 …

2. 0.5, 1, 1.5, 2 ...

3. –2/4, –1/4, 0, 1/4 ...

4. –6, –18, –54 ...Solutions to Worked Example 2:

5. Each term is 4 less than the previous term. So, the rule is: to subtract four from a term to get the next term. Therefore, the next two terms are 17 – 4 = 13, and 13 – 4 = 9. 29, 25, 21, 17, 13, 9 …

6. The sequence increases by 0.5 term to term. So, the next two terms are 2 + 0.5 = 2.5 and 2.5 + 0.5 = 3. 0.5, 1, 1.5, 2, 2.5, 3 ...

7. Each term is 1/4 of the size of the previous term. So, the next two terms are 1/4 + 1/4 = 2/4 and 2/4 + 1/4 = 3/4. –2/4, –1/4, 0, 1/4, 2/4, 3/4 ...

8. Each term is multiplied by 3. So, the next two terms are –54 × 3 = –162 and –162 × 3 = –486. –6, –18, –54, –162, –486 ...

9. Rules of Sequences

Let's look at two types of rules: the term-to-term rule and the position-to-term rule.

The term-to-term rule describes how to get from one term to the next. The position-to-term rule allows us to compute the value of any term. 3.1 Term-to-term rule

A term-to-term rule allows us to find the next number in the sequence if we know the previous term (or terms). To create such a rule:

Indicate the value that the sequence starts at. Provide a description of what is done to each term in order to get the next term in the sequence. Worked Example 3

What will the term-to-term rule be for 1, 3, 5, 7 ...? Solutions to Worked Example 3:

We can see the sequence goes up by 2 every time.

So, we can now describe the term-to-term rule: Start with 1 and add 2 to each term to get the next one.

Worked Example 4

Write down the term-to-term rule for each sequence.

80, 40, 20, 10, 5 … 1, 4, 16, 64, 256, 1 024 … 18, 6, 2, 2/3 ...Solutions to Worked Example 4:

1. Each term in the sequence is half the previous term. So, we can now describe the term-to-term rule: Start with 80 and divide each term by 2 to get the next one.

2. We can see the sequence goes up in multiples of 4. So, we can now describe the term-to-term rule: Start with 1 and multiply each term by 4 to get the next one.

3. Each term in the sequence is one-third of the previous term. So, we can now describe the term-to-term rule: Start with 18 and divide each term by 3 to get the next one.

3.2 Position-to-term rule

A position-to-term rule allows us to compute the value of any term by giving us a formula from which we can calculate any term.

Notation Each term is written as 𝑇𝑛, where 𝑛 represents the position of the term in the sequence.

So, for example, the first term is written 𝑇1 and the 12th term is written as 𝑇12.

Worked Example 5

In a sequence, 𝑇𝑛 = 2𝑛 + 1, calculate the value of:

The first three terms The 100th term.Solutions to Worked Example 5:

1. To find the first three terms, substitute the term number into the given formula:
   𝑇1 = 2(1) + 1 = 3 𝑇2 = 2(2) + 1 = 5 𝑇3 = 2(3) + 1 = 7

2. 𝑇100 = 2(100) + 1 = 201

3.3 Patterns

Sequences can be represented as geometrical patterns. Sequence questions sometimes come in the form of matchstick or dot patterns.

Worked Example 6

Each sequence is made up of a pattern of sticks. For each sequence:

Draw the next pattern. Draw a table and complete it for each pattern. Write down the term-to-term rule. Work out the number of sticks needed for the 10th pattern. 4. Rules for Square, Cube and Triangular Sequences

Among the many types of sequences, square, cube, and triangular numbers represent some of the most interesting patterns to explore. Let's delve into each type to understand their characteristics and see some examples.

Square Numbers A square number is a number that can be expressed as the product of an integer with itself. In other words, it's the area of a square with sides of integer length.

Formula: If 𝑛 is an integer, then the 𝑛th square number is given by 𝑛².

Examples:

1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 Sequence of Square Numbers: 1, 4, 9, 16, 25, ...

Cube Numbers: A cube number is a number that can be expressed as the product of an integer multiplied by itself twice. It represents the volume of a cube with sides of integer length.

Formula: If 𝑛 is an integer, then the 𝑛th cube number is given by 𝑛³.

Examples:

1³ = 1 2³ = 8 3³ = 27 4³ = 64 5³ = 125 Sequence of Cube Numbers: 1, 8, 27, 64, 125, ...

Triangular Numbers: A triangular number is a number that forms a triangle with a dot or a ball. It's the sum of the natural numbers up to a certain number. These numbers are visually represented as equilateral triangles.

Formula: If 𝑛 is a natural number, the 𝑛th triangular number is given by 𝑛(𝑛 + 1)/2.

Examples:

The 1st triangular number: 1(1 + 1)/2 The 2nd triangular number: 2(2 + 1)/2 The 3rd triangular number: 3(3 + 1)/2 The 4th triangular number: 4(4 + 1)/2 The 5th triangular number: 5(5 + 1)/2 Sequence of Triangular Numbers: 1, 3, 6, 10, 15, ...