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International Baccalaureate · MYP 1

Sequences

⌚ ~74 min ▢ 7 blocks ✓ 32 exercises ➣ Prerequisites : Whole Numbers and Operations, Number sequences and rules
I Generating a Sequence from a Rule
Definition — Numerical Sequence
A numerical sequence is a list of numbers that follows a specific rule.
  • The first number is called the 1\(^{\text{st}}\) term.
  • The second number is called the 2\(^{\text{nd}}\) term.
  • The third number is called the 3\(^{\text{rd}}\) term.
  • And so on.
Definition — A Sequence Given by Its First Term and Its Rule
A sequence can be given by two pieces of information:
  • its first term (the starting point);
  • the rule from one term to the next (also called a recursive rule), which tells how to get each new term from the term just before it.
With these two pieces of information, all the terms of the sequence can be found, step by step.
Example
A sequence is given by its first term and its rule from one term to the next:
  • The first term is 5.
  • The rule is: “add 3 to the previous term to get the next term”.
Find the first five terms of this sequence.

The sequence is built step by step:
  • 1\(^{\text{st}}\) term: The starting point is 5.
  • 2\(^{\text{nd}}\) term: Take the previous term (5) and add 3. \(\rightarrow 5 + 3 = \textbf{8}\)
  • 3\(^{\text{rd}}\) term: Take the previous term (8) and add 3. \(\rightarrow 8 + 3 = \textbf{11}\)
  • 4\(^{\text{th}}\) term: Take the previous term (11) and add 3. \(\rightarrow 11 + 3 = \textbf{14}\)
  • 5\(^{\text{th}}\) term: Take the previous term (14) and add 3. \(\rightarrow 14 + 3 = \textbf{17}\)
$$5 \textcolor{olive}{\xrightarrow{\;+3\;}} 8 \textcolor{olive}{\xrightarrow{\;+3\;}} 11 \textcolor{olive}{\xrightarrow{\;+3\;}} 14 \textcolor{olive}{\xrightarrow{\;+3\;}} 17$$
The first five terms are: \(5, 8, 11, 14, 17\).

Skills to practice
  • Continuing a Sequence from Its Table
  • Calculating the First Terms
II Recognising Arithmetic and Geometric Sequences
Definition — The Step of a Pattern
In some patterns the same number is added each time, or the same number is subtracted each time: that number is the step. It is the difference between two terms that are next to each other in the pattern.
Such a pattern is also called an arithmetic sequence, and its step the common difference.
Definition — Geometric Sequence
A geometric sequence is a list of numbers where each term is the previous term multiplied by the same number, the common ratio.
Dividing by \(2\) is multiplying by \(\frac{1}{2}\): in \(64, 32, 16, \dots\) each term is the previous term multiplied by \(\frac{1}{2}\), so the common ratio is \(\frac{1}{2}\).
Example
What is the 5\(^{\text{th}}\) term of this sequence?
position 1 2 3 4 5
term 2 4 8 16 ?

The 5\(^{\text{th}}\) term is \(32\), because each term is multiplied by \(2\).
Figure

Skills to practice
  • Identifying the Rule
  • Deciding Whether a Sequence Is Arithmetic or Geometric
  • Solving Problems with Arithmetic and Geometric Sequences
III Going further
Skills to practice
  • Reasoning and Problem-Solving
  • Solving Problems in Context
  • Quiz