Patterns, Sequences & Algebraic Thinking

Q: Why is it (n-1) and not n in the nth-term formula?

Because reaching term $n$ from term 1 takes $(n-1)$ jumps, not $n$ jumps. Using $n$ instead of $(n-1)$ is the most common error in these questions and gives you one term too many.

Q: What's the difference between square numbers, triangular numbers, and Fibonacci?

Square numbers follow $a_n=n^2$, triangular numbers follow $a_n=\dfrac{n(n+1)}{2}$, and Fibonacci numbers follow the recursive rule $a_n=a_{n-1}+a_{n-2}$ with no simple direct formula at this level.

Diagnose the pattern type first, then let the formula do the work — the IB MYP 2 way.

Growing dot and tile pattern next to an arithmetic sequence of numbers with arrows showing constant difference

Subject

Mathematics

Curriculum

IB MYP

Grade

MYP 2

Topic

Patterns, Sequences & Algebraic Thinking

Reading

6 min

Difficulty

Standard

Quick facts

Difficulty

★★☆☆☆

Assessed under

Criterion A & Criterion B

Most-tested skill

nth-term & pattern justification

Prerequisites

Basic algebra, integer operations

You'll learn

Arithmetic, geometric & quadratic pattern rules

Revision time

45–60 min

Every 'staircase' or 'dot pattern' question in IB MYP 2 Maths is really testing one skill first: can you diagnose the pattern type before reaching for a formula? Arithmetic sequences and geometric sequences look similar in a list of numbers, but the subtraction test and division test tell them apart instantly. Once you know the type, the nth-term formula lets you jump straight to Pattern 50 without drawing 49 pictures. This teaser walks through the five ideas examiners test most — arithmetic vs geometric sequences, nth-term formulas, reverse-engineering unknown terms, turning picture patterns into algebraic rules, and the named number patterns (square, triangular, Fibonacci). Master the $(n - 1)$ trap and the second-differences test here, then head to the full revision notes for every worked example, trap, and mock question.

What you’ll be able to do

✓Distinguish arithmetic from geometric sequences using the subtraction and division tests

✓Apply $a_n=a_1+(n-1)d$ and $a_n=a_1\times r^{n-1}$ correctly

✓Find $a_1$ and $d$ from any two given terms using simultaneous equations

✓Check whether a target value is actually a term in a sequence

✓Convert a growing picture pattern into a position-to-term algebraic rule

✓Recognise square numbers, triangular numbers and Fibonacci-style rules

✓Use second differences to spot a quadratic (non-linear) pattern

✓Avoid the classic $(n-1)$ substitution error under exam pressure

Arithmetic vs Geometric Sequences: Run the Test First

Before touching any formula, subtract consecutive terms: $a_{2} - a_{1}$ , then $a_{3} - a_{2}$ . If both match, it's arithmetic and that value is the common difference $d$ — even sequences like $20, 17, 14, 11, \dots$ with $d = - 3$ count. If the subtractions differ, try dividing instead: if $a_{2} \div a_{1}$ equals $a_{3} \div a_{2}$ , it's geometric with common ratio $r$ , which can grow, shrink, or alternate sign.

Two sequences side by side, one arithmetic with subtraction arrows and one geometric with division arrows

Feature	Arithmetic Sequence	Geometric Sequence
Constant value	Common difference, $d$	Common ratio, $r$
Found by	Subtracting consecutive terms	Dividing consecutive terms
nth-term formula	$a_{n} = a_{1} + (n - 1) d$	$a_{n} = a_{1} \times r^{n - 1}$

Exam tip

Never name a sequence arithmetic or geometric from just one subtraction or division — check two different pairs of terms before you commit.

Common mistake

Assuming the type after a single calculation, then applying the wrong formula when the pattern actually changes elsewhere.

Mini summary

Subtract first, divide second — the test you run determines which nth-term formula you're allowed to use.

nth-Term Formulas and the (n−1) Trap

Once the type is confirmed, $a_{n} = a_{1} + (n - 1) d$ generates any arithmetic term and $a_{n} = a_{1} \times r^{n - 1}$ does the same for geometric sequences. The single biggest scoring error is counting terms instead of jumps: getting from term 1 to term $n$ takes exactly $(n - 1)$ steps, never $n$ steps.

Number line showing term positions 1 to 10 with jump arrows counted as n-1

Exam tip

Write the formula with the actual numbers substituted in before simplifying — the method mark is awarded for correct substitution even if arithmetic afterwards slips.

Common mistake

Writing $a_{n} = a_{1} + n d$ (or $a_{1} \times r^{n}$ ) and computing one jump too many, landing on the wrong term entirely.

Mini summary

Count jumps, not terms: $(n - 1)$ is the number of steps from the first term to term $n$ .

Reverse-Engineering: Finding a₁ and d from Two Terms

Exam questions rarely give a full list — more often you get two specific terms, like the 4th and the 8th, and must reconstruct $a_{1}$ and $d$ . Write the nth-term formula twice, once per known position using $(n - 1)$ explicitly, then solve the pair of equations simultaneously. The same rearranged formula lets you check whether a target value is genuinely part of the sequence: solve for $n$ and confirm it's a positive whole number.

Two simultaneous equations derived from the 4th and 8th terms of a sequence, solved for a1 and d

Exam tip

Always write out '4th term means $n = 4$ , so the coefficient of $d$ is $(4 - 1) = 3$ ' before simplifying any equation.

Common mistake

Substituting $n$ directly instead of $(n - 1)$ when building the simultaneous equations, which gives the wrong $d$ and $a_{1}$ even with perfect algebra afterward.

Mini summary

Two known terms → two equations with $(n - 1)$ → solve simultaneously for $a_{1}$ and $d$ .

Turning Picture Patterns into Algebraic Rules

A growing matchstick, tile, or dot pattern is just an arithmetic sequence wearing a disguise. Count the constant amount added at each stage — that's $d$ — then build the rule $a_{n} = d n + (a_{1} - d)$ . A term-to-term rule only describes the next stage from the current one; the position-to-term rule gives any stage directly from $n$ , which is what 'general rule' means here.

Three stages of a matchstick square pattern with the constant increase highlighted and an algebraic rule written below

Exam tip

'Explain' questions need written justification using correct vocabulary (common difference, second difference, ratio) — a correct number with no explanation typically scores zero on the reasoning mark.

Common mistake

Using $a_{1}$ as the constant term instead of $(a_{1} - d)$ — always test $n = 1$ in your rule to catch this instantly.

Mini summary

Find $d$ from the picture, use $a_{n} = d n + (a_{1} - d)$ , and always verify with $n = 1$ .

Square, Triangular & Fibonacci Numbers

Not every pattern is arithmetic or geometric. When neither differences nor ratios stay constant, take the differences of the differences — the second differences. If those are constant, the pattern is quadratic, like square numbers $1, 4, 9, 16, \dots$ with $a_{n} = n^{2}$ . Triangular numbers $1, 3, 6, 10, \dots$ follow $a_{n} = \frac{n ( n + 1 )}{2}$ , while Fibonacci $1, 1, 2, 3, 5, 8, \dots$ has no simple direct formula at this level — each term is just the sum of the two before it.

Dot diagrams for square numbers and triangular numbers with first and second differences labelled underneath

Exam tip

If first differences aren't constant, don't give up — check second differences before assuming the pattern has no algebraic structure.

Mini summary

Square, triangular, and Fibonacci patterns show that not every sequence is arithmetic or geometric — second differences reveal quadratic patterns.

Quick formula sheet

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

nth term of an arithmetic sequence: first term plus

(n - 1)

lots of the common difference. — Count jumps, not terms — $(n-1)$ jumps get you from term 1 to term $n$.

a_{n} = a_{1} \times r^{n - 1}

nth term of a geometric sequence: first term multiplied by the common ratio raised to

(n - 1)

. — Same $(n-1)$ rule as arithmetic, just multiplying instead of adding.

d = \frac{a _{m} - a _{k}}{m - k}

Common difference between any two terms at positions

m

and

k

n = \frac{a _{n} - a _{1}}{d} + 1

Rearranged nth-term formula to find the position of a given value; a non-integer or negative result means it isn't in the sequence.

a_{n} = n^{2}

The nth square number, built from an

n \times n

grid of dots.

a_{n} = \frac{n ( n + 1 )}{2}

The nth triangular number, built by stacking

1, 2, 3, \dots

dots.

a_{n} = a_{n - 1} + a_{n - 2}

The Fibonacci rule: each term is the sum of the two immediately before it.

Practice questions

Easy

Is the sequence $5, 9, 13, 17, \dots$ arithmetic or geometric? State the constant value.
Find the 6th term of the arithmetic sequence with $a_{1} = 8$ and $d = 3$ .
Write the first four square numbers and the first four triangular numbers.

Medium

A geometric sequence begins $2, 6, 18, \dots$ . Find the 5th term.
A pattern uses 5 tiles for Pattern 1 and adds 4 tiles each stage. Write the position-to-term rule and find Pattern 15.
Determine whether 100 is a term in the arithmetic sequence with $a_{1} = 4$ and $d = 6$ .

Challenge

In an arithmetic sequence, the 3rd term is 14 and the 7th term is 34. Find $a_{1}$ and $d$ .
A sequence has first differences $3, 5, 7, 9$ . Show that the second differences are constant and explain what this tells you about the pattern type.
Compare an arithmetic sequence with $a_{1} = 7, d = 6$ to a geometric sequence with $a_{1} = 7, r = 2$ : state one similarity and one quantified difference between their growth after 5 terms.

Frequently asked questions

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

Subtract consecutive terms first. If the differences are equal, it's arithmetic. If they're not, divide consecutive terms instead — equal ratios mean it's geometric. Always check at least two pairs before deciding.

Why is it (n-1) and not n in the nth-term formula?+

Because reaching term $n$ from term 1 takes $(n - 1)$ jumps, not $n$ jumps. Using $n$ instead of $(n - 1)$ is the most common error in these questions and gives you one term too many.

How do I find the first term and common difference if I'm only given two terms?+

Write the nth-term formula twice, once for each given position, using $(n - 1)$ explicitly for each. Then solve the two equations simultaneously for $a_{1}$ and $d$ .

How do I turn a growing picture pattern into an algebraic rule?+

Find the constant amount added between stages — that's $d$ — then use $a_{n} = d n + (a_{1} - d)$ . Always test your rule at $n = 1$ to make sure the constant term is correct.

What's the difference between square numbers, triangular numbers, and Fibonacci?+

Square numbers follow $a_{n} = n^{2}$ , triangular numbers follow $a_{n} = \frac{n ( n + 1 )}{2}$ , and Fibonacci numbers follow the recursive rule $a_{n} = a_{n - 1} + a_{n - 2}$ with no simple direct formula at this level.

What do I do if a pattern isn't arithmetic or geometric?+

Check the second differences (the differences between the differences). If those are constant, the pattern is quadratic, like the square numbers.

Get the Full MYP 2 Patterns & Sequences Notes

Every worked example and exam trap explained step by step Full formula derivations for arithmetic, geometric and quadratic patterns Criterion A and B style mock questions with guidance Printable summary sheet for fast last-minute revision

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