IB MYP 3 Mathematics 3.3 Sequences, Patterns and Functions Study Notes - New Syllabus

3.3 – Sequences, Patterns and Functions

Sequences and patterns describe how numbers or quantities change from one term to the next. By identifying the rule of a pattern, we can find missing terms, predict future terms and describe the relationship between the term number and its value.

Sequences

A sequence is an ordered list of numbers that follows a particular pattern or rule. Each number in a sequence is called a term.

For example:

\(4,\ 7,\ 10,\ 13,\ 16,\ldots\)

The terms are:

• 1st term: \(4\)
• 2nd term: \(7\)
• 3rd term: \(10\)
• 4th term: \(13\)
• 5th term: \(16\)

The dots \(\ldots\) indicate that the sequence continues.

Finding the Pattern

To identify a sequence rule, compare consecutive terms and look for a consistent change.

Consider:

\(5,\ 9,\ 13,\ 17,\ 21,\ldots\)

Each term increases by \(4\).

\(5\rightarrow9\rightarrow13\rightarrow17\rightarrow21\)

Therefore, the term-to-term rule is:

Add \(4\).

 Arithmetic Sequences

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant.

The constant difference is called the common difference, usually written as \(d\).

Example:

\(12,\ 17,\ 22,\ 27,\ 32,\ldots\)

The common difference is:

\(d=17-12=5\)

So the sequence increases by \(5\) each time.

Decreasing arithmetic sequence:

\(30,\ 26,\ 22,\ 18,\ 14,\ldots\)

Here:

\(d=26-30=-4\)

Therefore, the sequence decreases by \(4\) each time.

Finding the Next Terms of an Arithmetic Sequence

Once the common difference is known, continue adding or subtracting it.

For example:

\(7,\ 11,\ 15,\ 19,\ldots\)

The common difference is \(4\).

Therefore:

\(19+4=23\)

\(23+4=27\)

The next two terms are \(23\) and \(27\).

Term Number and Position

The position of a term tells us where it appears in the sequence.

For an arithmetic sequence, the \(n\)th term can be found using:

\(\boxed{T_n=a+(n-1)d}\)

where:

• \(T_n\) = the \(n\)th term
• \(a\) = the first term
• \(n\) = the position of the term
• \(d\) = common difference

For the sequence \(5,8,11,14,\ldots\):

\(a=5,\quad d=3\)

Therefore:

\(T_n=5+(n-1)3\)

\(T_n=3n+2\)

The \(10\)th term is:

\(T_{10}=3(10)+2=32\)

Answer: \(32\)

Finding the Rule from the First Terms

A sequence can be represented by a rule involving its term number \(n\).

Consider:

\(4,\ 7,\ 10,\ 13,\ldots\)

The common difference is \(3\).

Using the arithmetic sequence formula:

\(T_n=4+(n-1)3\)

\(T_n=3n+1\)

Therefore, the position-to-term rule is:

\(T_n=3n+1\)

 Finding a Missing Term

If a term is missing, use the common difference or the nth-term rule.

Example:

\(6,\ 10,\ \square,\ 18,\ 22\)

The common difference is \(4\).

\(10+4=14\)

Therefore, the missing term is \(14\).

 Geometric Sequences

A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by the same number.

The constant multiplier is called the common ratio, written as \(r\).

Example:

\(3,\ 6,\ 12,\ 24,\ 48,\ldots\)

Each term is multiplied by \(2\).

\(r=\dfrac{6}{3}=2\)

Another example:

\(80,\ 40,\ 20,\ 10,\ldots\)

The common ratio is:

\(r=\dfrac{40}{80}=\dfrac{1}{2}\)

Note: An arithmetic sequence has a constant difference, while a geometric sequence has a constant ratio.

Patterns

Patterns can involve numbers, shapes, diagrams or other quantities. A pattern rule describes how the pattern changes.

For example, suppose a pattern has:

The number of tiles increases by \(3\) each time, so the rule is:

\(T_n=3n+1\)

This allows us to find the number of tiles in any figure without drawing all the previous figures.

Sequences as Functions

A sequence can be viewed as a function where the term number is the input and the term value is the output.

For example:

\(T_n=2n+3\)

For example, when \(n=8\):

\(T_8=2(8)+3=19\)

Answer: \(19\)