Sequences

Formula manipulation

Proportion

MYP Standard Mathematics Sequences Notes

Sequences Are Ordered Lists With Term Numbers

Definition

Sequence

An ordered list of numbers where each number in the list is called a term.

Because a sequence is ordered, the position matters.

We label terms using subscripts:

$u_1$ is the first term
$u_2$ is the second term
$u_3$ is the third term
in general, $u_n$ is the $n$th term

Here, the letter $u$ names the sequence, and the subscript tells you the term number.

Example

The sequence $$1, 3, 5, 7, 9, \ldots$$ has a clear first term, second term, third term, and so on.

Definition

Index (of a term)

Its subscript (for example, the index of $u_5$ is 5), which indicates the term’s position in the sequence.

Hint

Using $u_1, u_2, u_3, \ldots$ is more useful than labels like $a,b,c,\ldots$ because the subscript directly shows the position.
This matters when you want, for example, "the 20th term".

Note

Different subjects and contexts sometimes start sequences at $u_0$ instead of $u_1$ (especially in computer science).
Always check what "first term" means in the question.

Explicit Formulae Give A Direct Position-To-Term Rule

An explicit (position-to-term) formula lets you calculate any term without calculating all the ones before it.

For the odd numbers, $$u_n = 2n - 1 \quad \text{for } n \ge 1$$

Check it quickly:

$u_1 = 2(1)-1 = 1$
$u_2 = 2(2)-1 = 3$
$u_5 = 2(5)-1 = 9$

A key idea is that in sequences, $n$ is a term number, so it must be a whole number.

Common Mistake

In a sequence, $n$ represents a position (1st, 2nd, 3rd, …), so you do not substitute values like $n=2.5$.

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MYP Standard Mathematics Sequences Notes

Sequences Are Ordered Lists With Term Numbers

Explicit Formulae Give A Direct Position-To-Term Rule

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Sequences Are Ordered Lists With Term Numbers

Explicit Formulae Give A Direct Position-To-Term Rule

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Number and Algebra Foundations7 subtopics

Coordinate Geometry and Linear Modelling5 subtopics

Functions and Quadratic Relationships4 subtopics

Sets, Logic and Probability4 subtopics

Geometry of Shapes, Surface Area and Volume3 subtopics

Statistical Measures and Data Representation3 subtopics

Right Triangles and Circle Geometry3 subtopics

Sequences, Formulae and Proportional Reasoning3 subtopics

Sampling Methods and Bivariate Data2 subtopics

MYP Standard Mathematics Sequences Notes

Number and Algebra Foundations7 subtopics

Coordinate Geometry and Linear Modelling5 subtopics

Functions and Quadratic Relationships4 subtopics

Sets, Logic and Probability4 subtopics

Geometry of Shapes, Surface Area and Volume3 subtopics

Statistical Measures and Data Representation3 subtopics

Right Triangles and Circle Geometry3 subtopics

Sequences, Formulae and Proportional Reasoning3 subtopics

Sampling Methods and Bivariate Data2 subtopics

Sequences Are Ordered Lists With Term Numbers

Explicit Formulae Give A Direct Position-To-Term Rule

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Sequences Are Ordered Lists With Term Numbers

Explicit Formulae Give A Direct Position-To-Term Rule

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