What is a sequence in mathematics, and how is it relevant to the IB MYP curriculum?

In mathematics, a sequence is an ordered list of numbers following a specific pattern or rule. In the IB MYP curriculum, understanding sequences is crucial as it helps students develop skills in identifying patterns, predicting future terms, and applying mathematical reasoning. Sequences form the foundation for more advanced topics like series and calculus.

How do you find the nth term of an arithmetic sequence?

To find the nth term of an arithmetic sequence, use the formula: nth term = a + (n - 1) * d, where 'a' is the first term, 'd' is the common difference between consecutive terms, and 'n' is the term number. This formula helps students in the IB MYP Mathematics program to predict any term in the sequence without listing all the previous terms.

What distinguishes a geometric sequence from an arithmetic sequence?

A geometric sequence is characterized by a constant ratio between consecutive terms, while an arithmetic sequence has a constant difference. In a geometric sequence, each term is found by multiplying the previous term by a fixed number called the common ratio. Understanding these differences is essential for IB MYP students to analyze and solve problems involving different types of sequences.

Can you explain how to determine if a sequence is convergent or divergent?

A sequence is convergent if its terms approach a specific value as the term number increases indefinitely. Conversely, it is divergent if the terms do not approach a specific limit. In the IB MYP Mathematics curriculum, recognizing convergence or divergence helps students understand the behavior of sequences and their applications in real-world contexts.

What are some real-life applications of sequences that IB MYP students might encounter?

Sequences have numerous real-life applications, such as in finance for calculating interest, in computer science for algorithm design, and in nature for modeling growth patterns. For IB MYP students, understanding these applications enhances their ability to connect mathematical concepts with practical situations, fostering a deeper appreciation of the subject.

Why is "Writing $u_n = a + nd$ instead of $u_n = a + (n-1)d$." a common mistake in Sequences?

Why it happens: Slight formula error. How to avoid it: The first term is u_1 = a, not a + d. The formula has (n-1) — the FIRST term uses n=1 giving zero extra differences.

Why is "Treating a geometric sequence as arithmetic." a common mistake in Sequences?

Why it happens: Spotting the wrong pattern type. How to avoid it: Check: are consecutive terms a CONSTANT DIFFERENCE (AP) or CONSTANT RATIO (GP)? If neither, look for quadratic / other patterns.

Why is "Claiming a value is in the sequence when $n$ is non-integer." a common mistake in Sequences?

Why it happens: Forgetting that n must be a positive integer. How to avoid it: If your solve gives n = 4.7, the value is NOT in the sequence. Only integer n counts.

NumberIB MYP Maths Extended Level

Sequences

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Sequences (Number) — IB MYP Mathematics (Extended): patterns, arithmetic and geometric sequences

A sequence is an ordered list of numbers following a rule. This MYP Mathematics Extended note covers arithmetic and geometric sequences, the $n$ th term formula, and pattern investigations.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Identify arithmetic and geometric sequences.
MYP Mathematics A — Find any term using the $n$ th term formula.
MYP Mathematics B — Investigate a pattern and conjecture a rule.
MYP Mathematics C — Justify a general formula.

Arithmetic sequences

Constant difference between consecutive terms.

An arithmetic sequence has a CONSTANT difference between consecutive terms. Examples:

$3, 7, 11, 15, 19, \ldots$ (difference $d = 4$ ).
$20, 17, 14, 11, 8, \ldots$ (difference $d = -3$ — decreasing).

The general formula for the $n$ th term: $u_n = a + (n-1)d$ where $a$ is the FIRST term ( $u_1$ ) and $d$ is the common difference.

Worked example. The sequence $5, 9, 13, 17, \ldots$ — find the 20th term.

$a = 5$ , $d = 4$ .
$u_{20} = 5 + (20 - 1) \times 4 = 5 + 76 = 81$ .

Worked example. Find which term equals 100 in the sequence $7, 11, 15, 19, \ldots$ .

$a = 7$ , $d = 4$ . We want $u_n = 100$ .
$7 + (n-1) \times 4 = 100$ .
$(n-1) \times 4 = 93$ .
$n - 1 = 23.25$ .
$n = 24.25$ — NOT an integer, so 100 is NOT in this sequence.

(For 100 to be a term, $n$ must be a positive integer.)

An arithmetic sequence rises in equal jumps. The 'common difference' $d$ is the constant step size between consecutive terms.

Arithmetic: $u_n = a + (n-1)d$ .
Common difference $d$ = consecutive difference (positive OR negative).
Check $n$ is a positive integer when finding which term equals a given value.

Geometric sequences

Constant RATIO between consecutive terms.

A geometric sequence has a CONSTANT RATIO between consecutive terms. Examples:

$2, 6, 18, 54, \ldots$ (ratio $r = 3$ ).
$1, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \ldots$ (ratio $r = \tfrac{1}{2}$ ).
$100, -20, 4, -0.8, \ldots$ (ratio $r = -\tfrac{1}{5}$ — alternating signs).

The general formula for the $n$ th term: $u_n = a r^{n-1}$ where $a$ is the first term and $r$ is the common ratio.

To find $r$ : divide ANY term by the previous one.

Worked example. Sequence: $3, 12, 48, 192, \ldots$ . Find the 8th term.

$a = 3$ , $r = 12 / 3 = 4$ .
$u_8 = 3 \times 4^{8-1} = 3 \times 4^7 = 3 \times 16\,384 = 49\,152$ .

Geometric sequences GROW (or SHRINK) exponentially. They model things like:

Compound interest.
Bacterial growth.
Radioactive decay (with $r < 1$ ).
Bouncing ball heights (with $r < 1$ ).

Geometric: $u_n = a r^{n-1}$ .
Find $r$ : divide any term by the previous.
$|r| > 1$ : grows exponentially. $|r| < 1$ : shrinks.

Other patterns

Not everything is arithmetic or geometric.

Many MYP sequences don't fit either pattern exactly. Common examples:

Square numbers: $1, 4, 9, 16, 25, \ldots$ — $u_n = n^2$ .
Triangle numbers: $1, 3, 6, 10, 15, \ldots$ — $u_n = \dfrac{n(n+1)}{2}$ .
Cubes: $1, 8, 27, 64, \ldots$ — $u_n = n^3$ .
Fibonacci: $1, 1, 2, 3, 5, 8, 13, \ldots$ — each term is the sum of the previous two.
Quadratic sequences: general term is $u_n = an^2 + bn + c$ . Find by looking at SECOND differences.

For criterion B (Investigating patterns), the typical approach:

Spot the pattern in the first few terms.
Conjecture a formula.
Test on the next term to check.
Justify the formula (the harder Extended-level step).

A great example: counting handshakes in a room of $n$ people. The pattern is $\dfrac{n(n-1)}{2}$ — the same formula as triangle numbers (because each person shakes hands with $n-1$ others, and each handshake is counted twice).

Square: $n^2$ . Triangle: $n(n+1)/2$ . Cubes: $n^3$ .
Fibonacci: each term = sum of previous two.
Quadratic: constant SECOND differences.
Criterion B: spot → conjecture → test → justify.

How it’s examined

Criterion A: find specific terms. Criterion B: pattern investigation. Criterion C: justify a formula. Criterion D: model a real situation.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — Sequences

Step-by-step solutions to past-paper-style questions on sequences, written exactly the way a tutor would explain them at the board.

1Find the 50th term of an AP
Getting started• arithmetic
Question
Find the 50th term of the sequence $4, 7, 10, 13, \ldots$ .
Step-by-step solution
1. Step 1
  Arithmetic: $a = 4$ , $d = 3$ .
2. Step 2
  $u_n = a + (n-1)d$ . So $u_{50} = 4 + 49 \times 3 = 4 + 147 = 151$ .
Answer
$u_{50} = 151$
2Find the 10th term of a GP
Building confidence• geometric
Question
Find the 10th term of $2, 6, 18, 54, \ldots$ .
Step-by-step solution
1. Step 1
  Geometric: $a = 2$ , $r = 6/2 = 3$ .
2. Step 2
  $u_n = a r^{n-1}$ . So $u_{10} = 2 \times 3^9 = 2 \times 19\,683 = 39\,366$ .
Answer
$u_{10} = 39\,366$
3Find which term equals a value
Building confidence• solve
Question
Which term of $5, 11, 17, 23, \ldots$ equals 95?
Step-by-step solution
1. Step 1
  AP: $a = 5$ , $d = 6$ . We want $u_n = 95$ .
2. Step 2
  $5 + (n-1) \times 6 = 95 \Rightarrow (n-1) \times 6 = 90 \Rightarrow n - 1 = 15 \Rightarrow n = 16$ .
3. Step 3
  $n = 16$ is a positive integer, so 95 IS in the sequence.
Answer
$n = 16$ — the 16th term equals 95.
4Investigate a pattern
Stretch• criterion B
Question
The sequence is $1, 3, 6, 10, 15, \ldots$ (triangle numbers). Find a formula for the $n$ th term and use it to find the 100th term.
Step-by-step solution
1. Step 1
  First differences: 2, 3, 4, 5 (not constant — NOT arithmetic).
2. Step 2
  Second differences: 1, 1, 1, 1 (constant) — QUADRATIC sequence.
3. Step 3
  Conjecture: $u_n = \dfrac{n(n+1)}{2}$ . Test: $u_4 = \dfrac{4 \times 5}{2} = 10$ . ✓
4. Step 4
  $u_{100} = \dfrac{100 \times 101}{2} = 5050$ .
Answer
$u_n = \dfrac{n(n+1)}{2}$ . $u_{100} = 5050$ .

Key Definitions and Keywords — Sequences

Definitions to memorise and the exact keywords mark schemes credit for sequences answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Sequence
Examiner keyword
An ordered list of numbers (terms) following a rule.
Arithmetic sequence (AP)
Examiner keyword
A sequence with constant difference $d$ between consecutive terms.
Geometric sequence (GP)
Examiner keyword
A sequence with constant ratio $r$ between consecutive terms.
Common difference $d$
$d = u_{n+1} - u_n$ for an arithmetic sequence.
Common ratio $r$
$r = u_{n+1} / u_n$ for a geometric sequence.

Common Mistakes and Misconceptions — Sequences

The traps other students keep falling into on sequences questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Writing $u_n = a + nd$ instead of $u_n = a + (n-1)d$ .
Why it happens
Slight formula error.
How to avoid it
The first term is $u_1 = a$ , not $a + d$ . The formula has $(n-1)$ — the FIRST term uses $n=1$ giving zero extra differences.
✕Treating a geometric sequence as arithmetic.
Why it happens
Spotting the wrong pattern type.
How to avoid it
Check: are consecutive terms a CONSTANT DIFFERENCE (AP) or CONSTANT RATIO (GP)? If neither, look for quadratic / other patterns.
✕Claiming a value is in the sequence when $n$ is non-integer.
Why it happens
Forgetting that $n$ must be a positive integer.
How to avoid it
If your solve gives $n = 4.7$ , the value is NOT in the sequence. Only integer $n$ counts.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Sequences — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Sequences

Short Study Notes in the form of Slides

Short Study Notes — Sequences

Detailed Notes

What you’ll learn

How it’s examined

1Find the 50th term of an AP

2Find the 10th term of a GP

3Find which term equals a value

4Investigate a pattern

Sequence

Arithmetic sequence (AP)

Geometric sequence (GP)

Common difference ddd

Common ratio rrr

✕Writing un=a+ndu_n = a + ndun​=a+nd instead of un=a+(n−1)du_n = a + (n-1)dun​=a+(n−1)d.

✕Treating a geometric sequence as arithmetic.

✕Claiming a value is in the sequence when nnn is non-integer.

Practice questions

Past paper style quiz

Assess your understanding

Sequences — frequently asked questions

Common difference $d$

Common ratio $r$

✕Writing $u_n = a + nd$ instead of $u_n = a + (n-1)d$ .

✕Claiming a value is in the sequence when $n$ is non-integer.