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aiMOOC

Sequences including quadratic patterns - Mathematics Year 9

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Introduction

Does a pattern add the same amount, multiply by a constant ratio, or grow with changing first differences? Separate arithmetic, geometric and quadratic models before predicting later terms.

Classify six sequences and fully verify two proposed position-to-term rules. Connect numerical tables, an explicitly constructed dot pattern and algebra, and explain why a finite list alone does not prove a unique rule forever.

Construct and distinguish sequence models

A sequence is an ordered list of terms. A position-to-term rule gives the term at index n; here indices are positive integers starting at 1. A term-to-term rule instead tells how to move from one term to the next and needs a starting term. A term value and its index are different quantities: the fifth term need not equal 5.

An arithmetic sequence adds a constant first difference d. Its nth term is a+(n −1)d, where a is the first term. For 5, 8, 11, 14, 17, the difference is 3 and the rule 5+3(n −1)=3 n+2. For 20, 16, 12, 8, 4, the difference is −4 and rule 20−4(n −1)=24−4 n. These are abstract numerical sequences; negative future values are not interpreted as a literal negative count of objects.

For a quadratic model an²+bn+c with nonzero a, the second differences are constant 2 a. They are the differences of consecutive first differences. The source 2, 5, 10, 17, 26 has first differences 3, 5, 7, 9, then second differences 2, 2, 2. Under a quadratic model this gives a=1. A nonzero constant first difference instead indicates a linear arithmetic model.

Construct the source pattern. At stage n, make an n-by-n square of dots and add one separate extra dot. This declared construction has exactly n²+1 dots for every positive integer n. It supplies a general reason for the formula, not merely a guess from the first five numbers.

Original square-plus-one pattern
n=1
O
Extra:O

n=2
OO
OO
Extra:O

n=3
OOO
OOO
OOO
Extra:O
O denotes one dot.

The three stages contain 2, 5 and 10 dots. Going from the n-square to the next square adds a border of 2 n+1 dots while the single extra dot remains. This explains the successive odd first differences 3, 5, 7, 9. A table and a constructed geometric pattern provide different representations of the same declared model.

Do not infer this unique construction from an unrelated rods photograph or a short list alone. A finite sequence can fit many formulas with different future values. State that an arithmetic, quadratic or geometric model has been chosen, or use an explicit construction giving the rule for every permitted index. Check several positions to catch an incorrect coefficient or an index starting atzero.

Classify six arithmetic, quadratic and geometric rows

A blank ten-by-ten grid containing one hundred equal cells. — Use grids to count square arrangements. A10-by-10 square supplies100 cells;the n²+1 construction at n=10 needs one additional dot outside that square. The blank source is not already a101-dot pattern.

Six sequences under their stated model families. First differences subtract the previous term; second differences subtract the previous first difference. Exponent notation^ in the last two rules denotes integer powers. These classifications concern the declared full rules; finite initial data alone do not rule out other constructions.

Sequence	First five terms	First differences	Second differences	Model	Declared nth term
A	5, 8, 11, 14, 17	3, 3, 3, 3	0, 0, 0	Arithmetic	3 n+2
B	20, 16, 12, 8, 4	−4, −4, −4, −4	0, 0, 0	Arithmetic	24−4 n
C	2, 5, 10, 17, 26	3, 5, 7, 9	2, 2, 2	Quadratic	n²+1
D	3, 8, 15, 24, 35	5, 7, 9, 11	2, 2, 2	Quadratic	n²+2 n
E	3, 6, 12, 24, 48	3, 6, 12, 24	3, 6, 12	Geometric	3×2^(n −1)
F	81, 27, 9, 3, 1	−54, −18, −6, −2	36, 12, 4	Geometric	81×(1/3)^(n −1)

For E, the source 3, 6, 12, 24 is geometric with common ratio 2; the next term is 48. Its rule 3×2^(n −1) uses exponent 0 at the first term, giving 3×1=3. A geometric sequence multiplies by a fixed ratio, whereas an arithmetic sequence adds a fixed difference. E's first differences 3, 6, 12, 24 do not stay constant and its second differences do not either.

For F, the ratio between consecutive terms is 1/3: 81→27→9→3→1. Its next term is 1/3, not −1. The nth term 81×(1/3)^(n −1) retains exact fractions when later terms are not integers. A decreasing sequence need not be arithmetic; its direction alone does not identify the model.

The two arithmetic rows have second differences zero, because their first differences are constant. In the nonconstant quadratic rows, the second difference is 2. Constant second difference zero is not evidence of a genuine nonzero quadratic coefficient. Also check the ratio directly when a geometric model is claimed; nonzero terms are needed to calculate a consecutive-term ratio by division.

For original source C, n²+1 at n=5 gives 25+1=26. For D, n²+2 n at n=5 gives 25+10=35. A formula that fits the first term but fails the second is not a verified rule. Keep the entire finite-table verification and the model or construction argument explaining its general form.

The numerical tables do not depend on colour names or an animated sound track. Every term, difference, ratio and index is written explicitly. Use the image models as support and independently calculate from the declared values rather than estimating rods from screen pixels.

Derive two nth rules and test finite-pattern limits

Derive and verify two quadratic rules. In C and D, the nonzero constant second difference 2 suggests a leading term n² under the quadratic assumption. Subtract the square-number sequence 1, 4, 9, 16, 25 from the five terms to identify what remains.

For C 2, 5, 10, 17, 26, the residuals are 1, 1, 1, 1, 1. Add constant 1 to obtain $C_{n} = n^{2} + 1 (n = 1, 2, 3, \dots)$ . For D 3, 8, 15, 24, 35, the residuals are 2, 4, 6, 8, 10, which equal 2 n. This gives $D_{n} = n^{2} + 2 n (n = 1, 2, 3, \dots)$ . Do not confuse a constant residual with a changing one.

Position n	n squared	C: n²+1	D: n²+2 n
1	1	2	3
2	4	5	8
3	9	10	15
4	16	17	24
5	25	26	35
10	100	101	120

The first five rows check all supplied terms. The tenth-position row is a prediction from the declared models: C at 10 gives 101 and D gives 120. It is not a newly observed physical arrangement. The source fifth term C at 5 gives 26 is distinct from n 5 itself and from first difference 9 at that stage.

Compare Corbettmaths’ original quadratic-sequence example 6, 9, 14, 21, 30 with our different C and D rows. The visible source formula is n²+5, not our n²+1. Focus on the first/second-difference method and then derive each new residual independently. Original: corbettmaths, Quadratic Sequences Version 1 - Corbettmaths. The written explanations and native tasks work without sound or the external player.

First-error analysis. The rule 2 n fits C at n=1 but gives 4 at n=2, where the actual term is 5. The first error assumes the initial value 2 implies doubling the index without testing further positions. For E, the proposal 3 n gives 3, 6, 9 rather than 3, 6, 12, so it wrongly treats geometric growth as arithmetic. Identify the first mismatch instead of only saying “wrong formula”.

For an unfamiliar quadratic 2 n²−n+3, the first five values are 4, 9, 18, 31, 48. First differences 5, 9, 13, 17 yield constant second difference 4, matching twice the leading coefficient 2. At position 10, the value is 200−10+3=193. Compare generating directly from the rule with checking first and second differences.

Why finite fitting is limited. Add(n −1)(n −2)(n −3)(n −4)(n −5) to C's n²+1 formula. The extra product is zero at each of positions 1–5, so the first five terms still agree. At 6, it is 5×4×3×2×1=120, so this different rule gives 157 rather than C's 37. The declared square-plus-one construction or quadratic-model restriction selects the intended rule; the first five values alone do not prove uniqueness.

Retain all six classifications, two complete nth-rule derivations, the verification table, source geometric next term 48 and one unfamiliar transfer. Keep a corrected first mismatch and an actual later retrieval response. Do not invent observed construction accuracy or a future performance record from a table of model predictions.

Interactive Tasks

Quiz: Test Your Knowledge

Source 2, 5, 10, 17, 26: first differences? (3, 5, 7, 9) (!2, 2, 2, 2) (!3, 3, 3, 3) (!5, 10, 17, 26)

Those second differences are? (2, 2, 2) (!3, 5, 7) (!0, 0, 0) (!4, 4, 4)

Source C fifth term fromn²+1? (26) (!5) (!25) (!10)

D fifth term fromn²+2 n? (35) (!26) (!25) (!15)

3, 6, 12, 24 has which next term under ratio 2? (48) (!27) (!36) (!30)

81, 27, 9, 3, 1 continues under ratio 1/3 with? (1/3) (!−1) (!0) (!3)

5, 8, 11, 14, 17 has which declared model? (Arithmetic difference 3) (!Geometric ratio 3) (!Quadratic second difference 3) (!No possible rule)

C tenth term under its construction? (101) (!26) (!100) (!20)

Can five matching terms uniquely prove a rule without assumptions? (No; other rules may fit then differ) (!Yes; five always proves all indices) (!Only if all values are positive) (!Only if a video shows the table)

Unfamiliar 2 n²−n+3 at n=10? (193) (!203) (!190) (!103)

Memory Game

Arithmetic	Sequence adding a constant difference
Geometric	Sequence multiplying by a constant ratio
Quadratic	Declared rule with a nonzero squared-index coefficient
First difference	Consecutive term minus previous term
Second difference	Consecutive first-difference change
Index	Positive integer position in these models

Drag and Drop

3 n+2	Declared rule for 5, 8, 11, 14, 17
24−4 n	Declared rule for 20, 16, 12, 8, 4
n²+1	Declared rule for 2, 5, 10, 17, 26
n²+2 n	Declared rule for 3, 8, 15, 24, 35
3 times 2 to n −1	Geometric rule for 3, 6, 12, 24, 48

Crossword Puzzle

ARITHMETIC	Constant first-difference model
GEOMETRIC	Constant-ratio model
QUADRATIC	Nonzero constant-second-difference model
DIFFERENCE	Change calculated by subtraction
POSITION	Index of a term
RESIDUAL	Value left after subtracting a proposed leading pattern

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Calculate source differences: Find 3, 5, 7, 9 then 2, 2, 2 for C.
Count the constructed stages: Check square-plus-one stages 2, 5, 10 without reading rods as data.
Continue the source geometric list: Use ratio 2 to obtain 48.
Read an index: Explain fifth term 26 versus position 5.

Standard

Classify all six rows: Use first/second differences or constant ratios with the stated model.
Derive two nth rules: Subtractn² and identify constant 1 versus linear 2 n residual.
Verify both full tables: Check n 1–5 and predict n 10 using the declared rules.
Correct the first mismatch: Show why 2 n fails C at n=2 and 3 n fails E at n=3.

Advanced

Transfer a new quadratic: Generate 2 n²−n+3 and verify second difference 4 and tenth term 193.
Explain finite-data ambiguity: Evaluate the five-factor correction at 1–5 and at 6.
Compare generation methods: Use a formula, a recurrence and a constructed dot pattern with explicit assumptions.
Retain actual later work: Keep six models, two derivations, error correction and a real delayed response.

Learning Check

Six model classifications: Use the actual differences or ratios.
Two full nth-rule derivations: Subtractn² and verify each original position.
Geometric contrasts: Give next terms and distinguish repeated addition.
Construction and first-error checks: Count actual schematic dots and correct 2 n for C.
Unfamiliar quadratic and ambiguity: Evaluate the new quadratic and finite-fitting counterexample.
Portfolio and evidence limits: Retain models, reasoning and actual later work.

Worked checks and assessment guidance

A arithmetic difference 3, rule 3 n+2; B arithmetic difference −4, rule 24−4 n; C quadratic second difference 2, n²+1; D quadratic second difference 2, n²+2 n; E geometric ratio 2, 3×2^(n −1); F geometric ratio 1/3, 81×(1/3)^(n −1). Model assumptions are declared, not uniquely inferred from finite data.
C residuals 1, 1, 1, 1, 1 give n²+1; D residuals 2, 4, 6, 8, 10 give n²+2 n. Both agree at 1–5; position 10 gives 101 and 120. The source C fifth term is 26.
Source 3, 6, 12, 24 multiplies 2, next 48.81, 27, 9, 3, 1 multiplies 1/3, next 1/3. Their first/second differences are not constant like the corresponding arithmetic/quadratic rows.3 n fails E at n=3.
Stage 1 square 1 plus extra 1=2; stage 2 square 4 plus 1=5; stage 3 square 9 plus 1=10. Rule 2 n matches only the first term before failing at n 2 with 4 instead 5. The declared general square-plus-one construction justifies n²+1.
2 n²−n+3 gives 4, 9, 18, 31, 48; first differences 5, 9, 13, 17; second difference 4; tenth 193. Addingproduct(n −1)…(n −5) toC leaves the first 5 terms but at 6 adds 120, giving 157 rather than 37. State the chosen family/construction.
Keep six classifications, two complete derivations, all verification positions, source geometric 48, first-error correction and unfamiliar transfer. Numerical predictions are not observed physical counts or evidence of unobserved delayed mastery.

Learning evidence

Retain six model classifications, two fully verified nth rules, source geometric next term, a general dot construction, first-error correction and unfamiliar transfer.

OER and Wikipedia

Department for Education: England mathematics programme of study, Key Stage 3, number, ratio, fluency, reasoning and problem solving. Year 9 allocation is editorial within KS3. Numerical models, tasks and feedback are original; images retain actual Commons attribution and videos name their original publisher. Reviewed 12 November 2026.

Cuisenaire-Rods.jpg, Celcom at en.wikipedia, CC BY 3.0. Unchanged original; caption and learning activities are original.

100 grid.PNG, The original uploader was Leightwing at English Wikiversity., Public domain. Unchanged original; caption and learning activities are original.