Patterns, Sequences & Algebraic Thinking

Q: How do I tell if a sequence is arithmetic or geometric?

Subtract consecutive terms first — if the result is always the same, it's arithmetic and that value is $d$. If subtracting gives different values, divide consecutive terms instead; a constant result there means it's geometric and that value is $r$.

Q: What's the difference between $u_n = u_1 + (n-1)d$ and $u_n = u_1 \cdot r^{n-1}$?

The first formula is for arithmetic sequences (adding $d$ repeatedly) and the second is for geometric sequences (multiplying by $r$ repeatedly). Use whichever matches your test result from subtracting or dividing terms.

Q: Why does my nth-term rule need to be checked at $n=1$?

Because the coefficient of $n$ only gives you the common difference — it doesn't guarantee the starting point is correct. Substituting $n=1$ confirms your constant term actually matches the real first term.

Q: How do I answer 'justify whether the student is correct' questions?

Calculate the actual value yourself, write it down, and then explicitly compare it in a sentence to the student's claim. A verdict alone, without the written calculation and comparison, usually only earns partial credit.

Q: Is a real-life pattern always exactly arithmetic or geometric?

Not forever — most real situations only follow the pattern for a while before a specific, named factor in the context (like a wall, a bill, or reduced income) causes it to break down.

Q: What do I do if the first differences of a sequence aren't constant?

Don't assume there's no pattern — calculate the differences of the differences (second differences). A constant second difference points to patterns like square or triangular numbers rather than a simple linear rule.

Crack arithmetic and geometric sequences, nth-term rules and real-world patterns for IB MYP 3 Maths

Number sequence blocks showing an arithmetic pattern increasing by a fixed amount

Subject

Mathematics

Curriculum

IB MYP

Grade

MYP 3

Topic

Patterns, Sequences & Algebraic Thinking

Reading

7 min

Difficulty

Standard

Quick facts

Difficulty

★★★☆☆

Exam weight

Core strand — Criterion B & C, every unit test + eAssessment

Prerequisites

Basic algebra, integer operations, substitution

You'll learn

nth-term rules for arithmetic & geometric sequences

Revision time

25-30 min

Every number pattern you meet in IB MYP 3 Maths is really one of two behaviours in disguise: adding the same amount each time, or multiplying by the same amount each time. Once you can tell arithmetic sequences and geometric sequences apart, the nth-term formulas practically write themselves — and suddenly 'find term 50' takes seconds instead of a long list. This topic sits at the heart of MYP Criterion B (Investigating Patterns) and Criterion C (Communicating), showing up in dot patterns, seating plans, savings problems and the on-screen eAssessment. This teaser walks through the five ideas that matter most: telling sequence types apart, using the two core formulas, building your own general rule, reading real-life word problems, and generalising patterns algebraically. For the full worked examples, common-mistake breakdowns and practice sets, the complete revision note is linked below.

What you’ll be able to do

✓Distinguish arithmetic sequences from geometric sequences by testing differences and ratios

✓Apply $u_n = u_1 + (n-1)d$ to find any term of an arithmetic sequence

✓Apply $u_n = u_1 \cdot r^{n-1}$ to find any term of a geometric sequence

✓Build a general linear nth-term rule from a common difference and first term

✓Check a draft nth-term rule against $n=1$ and a second term for accuracy

✓Identify whether a real-life context models an arithmetic or geometric pattern

✓Explain, with a named constraint, why a real-world pattern might break down

✓Justify claims about sequence terms using written calculated comparisons

Arithmetic vs Geometric Sequences

An arithmetic sequence adds the same fixed number, $d$ , every step; a geometric sequence multiplies by the same fixed factor, $r$ , every step. Test for arithmetic by subtracting consecutive terms — if the result repeats, that's $d$ . Test for geometric by dividing consecutive terms (later ÷ earlier) — if that repeats, that's $r$ . Plotted as $(n, u_{n})$ , arithmetic sequences sit on a straight line, while geometric sequences curve, steepening for growth ( $r > 1$ ) or flattening toward zero for decay ( $0 < r < 1$ ).

Comparison of a straight-line arithmetic sequence graph and a curved geometric sequence graph

Feature	Arithmetic sequence	Geometric sequence
Change per step	Add a fixed $d$	Multiply by a fixed $r$
Test	Subtract consecutive terms	Divide consecutive terms
Graph shape	Straight line	Curve (growth or decay)

Exam tip

Always confirm the same difference (or ratio) holds for at least two consecutive pairs before you commit to $d$ or $r$ .

Common mistake

Subtracting terms in the wrong order, or dividing consecutive terms of an arithmetic sequence looking for a ratio that isn't there.

Mini summary

Add = arithmetic; multiply = geometric — subtract first, divide second, and check twice.

The Two Core nth-Term Formulas

Once you know a sequence's type, one formula lets you jump straight to any term without listing every value in between. For arithmetic sequences use $u_{n} = u_{1} + (n - 1) d$ ; for geometric sequences use $u_{n} = u_{1} \cdot r^{n - 1}$ . If you're only given two terms rather than the first term and the common difference, set up two equations and solve for $d$ or $r$ by comparing the gap between the term positions, not the term numbers themselves.

Formula card showing the arithmetic nth-term formula and geometric nth-term formula side by side

Exam tip

'State the formula' is often its own mark — write $u_{n} = u_{1} + (n - 1) d$ symbolically before substituting any numbers.

Common mistake

Dividing the difference between two given terms by the term numbers (e.g. 7 or 3) instead of by the number of steps between them (e.g. 4).

Mini summary

$u_{n} = u_{1} + (n - 1) d$ for add-patterns; $u_{n} = u_{1} \cdot r^{n - 1}$ for multiply-patterns.

Building Your Own General Rule

An nth-term rule lets you find any term directly from its position $n$ — essential once a question asks for 'term 50'. For a linear pattern, the coefficient of $n$ is always the common difference $d$ ; the constant term is whatever makes the rule true when $n = 1$ . Always double-check your finished rule against a term you did NOT use to build it — that safety check is exactly what examiners look for.

Dot pattern arrangement with the corresponding nth-term rule written underneath

Exam tip

Substitute $n = 1$ into your draft rule immediately; if it doesn't exactly match the real first term, adjust the constant using $u_{n} = d n + (u_{1} - d)$ .

Common mistake

Writing the rule as just $d n$ (e.g. $3 n$ ) with no correction constant, because the coefficient looks right but the starting point was never checked.

Mini summary

Coefficient of $n$ = common difference; constant = whatever makes $n = 1$ correct — then verify with a second term.

Spotting Patterns in Real-Life Word Problems

Most 'real-life' exam questions are simply an arithmetic or geometric sequence wearing a costume. Fixed-amount contexts — a flat fee plus a constant add-on, like seat rows or weekly savings — are arithmetic. Fixed-percentage contexts — interest rates, doubling or halving — are geometric. When asked why a model might not hold forever, always name a concrete, context-specific limiting factor, never a vague generalisation.

Theatre seating rows increasing arithmetically, with a wall symbol showing a physical limit

Exam tip

For 'explain one real-world reason' questions, give ONE strong, specific, named constraint rather than two weak vague ones.

Common mistake

Assuming a real-world pattern continues forever with the exact same difference or ratio, e.g. claiming a theatre has unlimited seats or a bank balance grows without limit.

Mini summary

Flat add-on = arithmetic; fixed percentage = geometric; always name a real limiting factor.

Generalising Patterns Algebraically

Generalising a pattern means replacing the specific numbers you counted with one algebraic expression in $n$ that works for every term at once — the bridge between noticing a pattern and being able to use or prove it. Not every pattern is linear: sequences like squares or triangular numbers don't have a constant first difference, so you need to look at the differences of the differences to spot the underlying structure. Spotting whether a pattern is linear (constant first difference) or needs a second look tells you which type of rule to search for.

Sequence of square number dot patterns with first and second differences labelled underneath

Exam tip

If the first differences aren't constant, calculate the second differences before assuming there's no pattern at all.

Common mistake

Giving up on a pattern as 'random' as soon as the first differences aren't equal, instead of checking the differences of the differences.

Mini summary

Constant first difference → linear rule; unequal first differences → check second differences before generalising.

Quick formula sheet

u_{n} = u_{1} + (n - 1) d

nth term of an arithmetic sequence, using first term

u_{1}

and common difference

d

. — Start at $u_1$, then take $(n-1)$ steps of size $d$.

d = u_{n} - u_{n - 1}

Finding the common difference from any two consecutive terms. — Later minus earlier, always.

u_{n} = u_{1} \cdot r^{n - 1}

nth term of a geometric sequence, using first term

u_{1}

and common ratio

r

. — Start at $u_1$, then multiply by $r$, $(n-1)$ times.

r = \frac{u _{n}}{u _{n - 1}}

Finding the common ratio from any two consecutive terms. — Later over earlier — division, not subtraction.

u_{n} = d n + (u_{1} - d)

General linear nth-term rule built directly from the common difference and first term. — Coefficient of $n$ is $d$; the rest is whatever fixes $n=1$.

Practice questions

Easy

State whether 5, 9, 13, 17 is arithmetic or geometric and give the common difference or ratio.
Find the common ratio of the sequence 3, 6, 12, 24.
Write the nth-term rule for the sequence 2, 5, 8, 11.

Medium

An arithmetic sequence has $u_{2} = 9$ and $u_{5} = 21$ . Find the common difference $d$ .
A savings pattern starts at $10 and increases by $4 each week. Predict the saving in week 6 and state the type of sequence.
Check whether the rule $u_{n} = 3 n + 1$ correctly gives the first term of the sequence 4, 7, 10, 13.

Challenge

A geometric sequence has $u_{1} = 5$ and $u_{4} = 135$ . Find the common ratio $r$ .
A theatre's row pattern is arithmetic with $a = 8$ , $d = 3$ . Explain one specific reason the model might not hold for row 20 in a real building.
A sequence of square numbers is 1, 4, 9, 16, 25. Calculate the first and second differences and explain what they show about the pattern.

Frequently asked questions

How do I tell if a sequence is arithmetic or geometric?+

Subtract consecutive terms first — if the result is always the same, it's arithmetic and that value is $d$ . If subtracting gives different values, divide consecutive terms instead; a constant result there means it's geometric and that value is $r$ .

What's the difference between $u_n = u_1 + (n-1)d$ and $u_n = u_1 \cdot r^{n-1}$?+

The first formula is for arithmetic sequences (adding $d$ repeatedly) and the second is for geometric sequences (multiplying by $r$ repeatedly). Use whichever matches your test result from subtracting or dividing terms.

Why does my nth-term rule need to be checked at $n=1$?+

Because the coefficient of $n$ only gives you the common difference — it doesn't guarantee the starting point is correct. Substituting $n = 1$ confirms your constant term actually matches the real first term.

How do I answer 'justify whether the student is correct' questions?+

Calculate the actual value yourself, write it down, and then explicitly compare it in a sentence to the student's claim. A verdict alone, without the written calculation and comparison, usually only earns partial credit.

Is a real-life pattern always exactly arithmetic or geometric?+

Not forever — most real situations only follow the pattern for a while before a specific, named factor in the context (like a wall, a bill, or reduced income) causes it to break down.

What do I do if the first differences of a sequence aren't constant?+

Don't assume there's no pattern — calculate the differences of the differences (second differences). A constant second difference points to patterns like square or triangular numbers rather than a simple linear rule.

Get the Full MYP 3 Patterns & Sequences Notes

Complete worked examples for arithmetic and geometric sequences, including Criterion B and C style justify/explain questions Step-by-step guidance on building and checking nth-term rules from scratch Full breakdown of common mistakes with fixes, plus original mock papers and exam-style practice questions for MYP 3 Maths

Get the Patterns, Sequences & Algebraic Thinking notes on RevisionPrep →

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