Non-linear relationships

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Find a constant difference
Use an nth-term rule and test it

Vocabulary: arithmetic sequence · common difference · term · position

Retrieve previous learning

Find the next two terms after 5, 8, 11.

Check: 14 and 17, because each term increases by 3.

Learn the concept

An arithmetic sequence adds the same difference d between consecutive terms. For 5, 8, 11, 14, the difference is 3, so the term at position n is aₙ = 5 + 3(n − 1) = 3n + 2. Starting from the first term and making n − 1 jumps explains the formula.

The tenth term is 5 + 3 × 9 = 32. A rule 'add three' predicts only the next term, while aₙ = 3n + 2 directly finds any position. Check position 1: 3 + 2 = 5.

Worked model

Subtract consecutive terms to establish d = 3.
Write first term plus (n − 1) differences: 5 + 3(n − 1).
Simplify to 3n + 2 and verify terms 1 and 4.

Guided practice

Find aₙ for 7, 11, 15, 19 and calculate a₂₀.

Hint: There are n − 1 jumps of 4 from the first term.

Check the guided answer

aₙ = 7 + 4(n − 1) = 4n + 3; a₂₀ = 83.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The nth term is always n times the common difference.”

Correction: Adjust for the first term: aₙ = a₁ + (n − 1)d.

Extend and reflect

Find the arithmetic sequence with a₅ = 14 and difference 3.

Guidance: a₁ = 14 − 4 × 3 = 2, so its nth term is 3n − 1.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Find a constant difference. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Generalise an arithmetic pattern from separated terms

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Recover a difference from separated positions
Form and justify a direct term rule

Vocabulary: term-to-term · nth term · constant difference · intercept

Retrieve previous learning

If the fifth term is 20 and each step adds 4, what is the fourth?

Check: 16, by subtracting one step of 4.

Learn the concept

Suppose a₂ = 9 and a₅ = 21. Three position steps increase the term by 12, so d = 12/3 = 4. Work backward from a₂ to a₁ = 5. Thus aₙ = 5 + 4(n − 1) = 4n + 1.

Check both supplied constraints: at n = 2 the rule gives 9; at n = 5 it gives 21. Two terms with their positions determine an arithmetic rule only because the constant-difference assumption has been stated. Without it, many other patterns could fit.

Worked model

Count 5 − 2 = 3 position gaps.
Divide change 21 − 9 = 12 by three to obtain d = 4.
Find a₁ = 9 − 4 = 5, then aₙ = 4n + 1.
Substitute n = 2 and n = 5 to check.

Guided practice

For an arithmetic sequence, a₃ = 14 and a₇ = 30. Find aₙ.

Hint: Four gaps account for an increase of 16.

Check the guided answer

d = 16/4 = 4; a₁ = 14 − 2 × 4 = 6, so aₙ = 6 + 4(n − 1) = 4n + 2.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“a₅ − a₂ is the one-step common difference.”

Correction: The difference spans three gaps, so divide by three.

Extend and reflect

If aₚ = u and a_q = v for p ≠ q, express d.

Guidance: For an arithmetic sequence, d = (v − u)/(q − p), then aₙ = u + (n − p)d.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Recover a difference from separated positions. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Describe repeated multiplication in a geometric sequence

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Identify a constant ratio
Calculate terms by repeated multiplication

Vocabulary: geometric sequence · common ratio · growth · decay

Retrieve previous learning

How does 3 become 12 by multiplication?

Check: Multiply by 4.

Learn the concept

A geometric sequence multiplies by the same nonzero ratio r at each step. In 3, 6, 12, 24, r = 2. Its nth term is aₙ = 3 × 2^(n−1): n = 1 has zero multiplications, n = 4 has three.

A ratio between 0 and 1 creates decreasing positive terms, such as 80, 40, 20, 10 with r = 1/2. A negative ratio alternates signs, for example 2, −6, 18, −54 has r = −3. Difference and ratio are different tests.

Worked model

Divide each term by its predecessor: 6/3 = 2, 12/6 = 2.
Use a₁ = 3 and r = 2 in aₙ = a₁r^(n−1).
Calculate a₅ = 3 × 2⁴ = 48.
Verify the recurrence: 24 × 2 = 48.

Guided practice

Find the ratio and fifth term of 81, 27, 9, 3, ...

Hint: Divide the second term by the first.

Check the guided answer

r = 27/81 = 1/3. The next term is 1, so the fifth term is 1/3; equivalently 81(1/3)^4 = 1/3.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Any sequence with increasingly large terms is geometric.”

Correction: Only a constant multiplicative ratio establishes a geometric sequence.

Extend and reflect

When r = 1, what happens to a nonzero geometric sequence?

Guidance: Every term equals a₁; its difference is also zero, so this special case is both arithmetic and geometric.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Identify a constant ratio. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Test whether a sequence has a fixed ratio

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Compare successive ratios accurately
Distinguish geometric patterns from lookalikes

Vocabulary: successive ratio · zero term · alternating sign · classification

Retrieve previous learning

What are 12/6 and 24/12?

Check: Both are 2.

Learn the concept

To test 6, 18, 54, 162, divide each later term by the previous term: 3, 3, 3. A fixed ratio confirms this finite list is consistent with a geometric rule aₙ = 6 × 3^(n−1). Check every available adjacent pair, not only the first.

For 2, 4, 8, 15, the first two ratios are 2 but 15/8 is not 2, so the displayed sequence is not geometric. Ratios cannot be formed by division through a zero term. State any special zero case explicitly instead of dividing by zero.

Worked model

For 4, −8, 16, −32, calculate −8/4 = −2.
Check 16/(−8) = −2 and −32/16 = −2.
Conclude the displayed terms have common ratio −2.
Predict the next term as 64, with a sign change.

Guided practice

Classify 9, 3, 1, 1/3.

Hint: Test each consecutive quotient.

Check the guided answer

3/9 = 1/3, 1/3 = 1/3 and (1/3)/1 = 1/3. It is geometric with r = 1/3.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Two consecutive equal ratios prove an entire supplied list geometric.”

Correction: Check every adjacent ratio in the supplied list and state the rule as an assumption for future terms.

Extend and reflect

What ratio would turn 8 into 0, and could the preceding 8, 4, 2 share it?

Guidance: The final ratio is 0, which differs from 1/2; a single fixed ratio cannot explain all four terms.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Compare successive ratios accurately. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Plot geometric terms against position

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Plot (position, term) pairs
Explain why exponential-looking points are discrete

Vocabulary: discrete point · position · exponential growth · vertical scale

Retrieve previous learning

For 2, 4, 8, what are the first three (position, term) pairs?

Check: (1, 2), (2, 4), (3, 8).

Learn the concept

For aₙ = 2 × 2^(n−1), plot (1, 2), (2, 4), (3, 8), (4, 16), (5, 32). The vertical jumps grow because each term doubles. The horizontal position is the term number n, not elapsed distance unless a context defines it.

Term positions are positive integers, so the sequence graph is a set of separate points. A smooth curve through them would represent an additional continuous function, not the sequence alone. Choose a vertical scale that includes 32 and label the units to avoid a misleading shape.

Worked model

Make a table for n = 1 through 5.
Calculate terms 2, 4, 8, 16, 32.
Plot five separate points using n on the horizontal axis.
Compare the vertical differences 2, 4, 8, 16 and the fixed ratio 2.

Guided practice

Plot the first four terms of 40, 20, 10, 5 as (n, aₙ). Describe the pattern.

Hint: n starts at 1.

Check the guided answer

Plot (1, 40), (2, 20), (3, 10), (4, 5). Each y-value halves; the points descend but remain above zero.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The horizontal spacing between terms should widen as the terms grow.”

Correction: Successive positions differ by 1; growth changes the vertical coordinate, not the term-number interval.

Extend and reflect

Compare graphs for ratios 2 and 1/2 with the same first term 8.

Guidance: Both have integer x-values; one rises 8, 16, 32 while the other falls 8, 4, 2 and approaches zero without reaching it.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Plot (position, term) pairs. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Build square, triangular and cube sequences

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Generate special sequences from definitions
Explain the geometry or arithmetic behind a term

Vocabulary: square number · triangular number · cube number · nth term

Retrieve previous learning

How many unit squares fit in a 4 by 4 square?

Check: 16, because 4 × 4 = 4².

Learn the concept

Square numbers count unit tiles in n by n squares: n² gives 1, 4, 9, 16, 25. Cube numbers count unit blocks in n by n by n cubes: n³ gives 1, 8, 27, 64, 125. Their growing differences show they are not arithmetic.

Triangular numbers add rows of 1, 2, ..., n objects. Pair two identical staircases into an n by (n + 1) rectangle, so one has Tₙ = n(n + 1)/2 objects. For n = 5, T₅ = 5 × 6/2 = 15.

Worked model

List square terms as 1², 2², 3², 4².
List triangular terms as 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4.
Use T₄ = 4 × 5/2 = 10.
List cube terms as 1³, 2³, 3³, 4³.

Guided practice

Find the sixth triangular number without drawing all rows.

Hint: Use n(n + 1)/2.

Check the guided answer

T₆ = 6 × 7/2 = 21, equal to 1 + 2 + 3 + 4 + 5 + 6.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The triangular nth term is n²/2.”

Correction: Two staircases form an n by (n + 1) rectangle, so Tₙ = n(n + 1)/2.

Extend and reflect

Why does Tₙ − Tₙ₋₁ = n?

Guidance: Each new triangular array adds one row of n objects; subtract the formulas to confirm.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Generate special sequences from definitions. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Recognise a special sequence and justify the rule

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Test square, triangular and cube hypotheses
Use differences and exact term values to reject a false pattern

Vocabulary: first difference · second difference · counterexample · hypothesis

Retrieve previous learning

What are 3² and 3³?

Check: 9 and 27.

Learn the concept

The sequence 1, 3, 6, 10, 15 has differences 2, 3, 4, 5, matching triangular numbers. The sequence 1, 4, 9, 16, 25 has differences 3, 5, 7, 9, matching squares. Both have constant second difference 1 or 2 respectively, yet their formulas differ.

A few displayed values do not prove a unique future rule. To classify a proposed special sequence, match a definition or structural pattern and test several positions. For 1, 8, 27, 64, position n has n³, so the fifth expected term is 125.

Worked model

For 1, 3, 6, 10, take differences 2, 3, 4.
Observe each new difference increases by 1.
Check T₄ = 4 × 5/2 = 10.
Predict T₅ = 15 under the triangular rule.

Guided practice

Classify 1, 4, 9, 16 and predict its sixth term.

Hint: Compare each term with n².

Check the guided answer

They are n² for positions 1–4; the sixth square under this rule is 6² = 36.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A constant second difference alone identifies square numbers.”

Correction: It indicates a quadratic pattern, but triangular numbers and many other quadratic sequences also have constant second differences.

Extend and reflect

What is the second difference for n² and for n(n + 1)/2?

Guidance: For n² it is 2; for triangular numbers it is 1.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Test square, triangular and cube hypotheses. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Graph special sequences with a table and digital tool

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Generate a reliable value table
Select an appropriate graph window and interpret discrete points

Vocabulary: table · scatter plot · window · discrete domain

Retrieve previous learning

Calculate T₄ = 4(5)/2.

Check: 10.

Learn the concept

Enter integer positions n = 1 to 5 and formulas n², n³ and n(n + 1)/2 into a spreadsheet or graphing table. The fifth values are 25, 125 and 15. A scatter plot with n on the horizontal axis shows separate points because term numbers are integers.

Use a vertical scale reaching at least 125 to show all three fifth terms; a smaller window can hide the cube point. Validate one row manually, such as n = 4 giving 16, 64 and 10, because a misplaced bracket in a formula can produce plausible but wrong values.

Worked model

Create a first column of n = 1, 2, 3, 4, 5.
Calculate n², n³ and n(n + 1)/2 in adjacent columns.
Check row n = 4 manually as 16, 64, 10.
Plot separate points with a y-range including 125 and label each series.

Guided practice

A digital plot omits the point (5, 125) for n³ because y-max is 100. What should change?

Hint: Check the point by hand before adjusting the window.

Check the guided answer

5³ = 125, so the point is valid but outside the current display. Set y-max above 125, for example 130.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A digital curve through the dots is the sequence itself.”

Correction: The sequence is defined only at integer positions; the smooth curve is an optional continuous extension.

Extend and reflect

How can you graph Tₙ and n² on one window without hiding their differences?

Guidance: Use the same labelled axes and a range covering the largest term; compare paired points at equal n.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Generate a reliable value table. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

State what can and cannot be inferred from early terms

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Extend a stated rule
Explain why finite observations alone do not determine a unique continuation

Vocabulary: extrapolate · conjecture · counterexample · rule

Retrieve previous learning

If aₙ = n², what is a₅?

Check: 25.

Learn the concept

If a rule is given, extrapolation is justified by it: for aₙ = n², a₁₀ = 100. If only 1, 4, 9, 16 are shown, 25 is a sensible square-number prediction but not logically forced. A rule could match four terms and change afterwards.

For example bₙ = n² + (n−1)(n−2)(n−3)(n−4) equals n² at n = 1, 2, 3, 4 because one factor is zero each time. At n = 5 the added product is 4 × 3 × 2 × 1 = 24, giving b₅ = 49. Structural evidence or an explicit definition is needed to justify the square continuation.

Worked model

Observe 1, 4, 9, 16 and conjecture n².
Under that rule calculate fifth term 25.
Construct bₙ by adding a product zero at positions 1–4.
At position 5 obtain 49, proving the four terms do not uniquely fix a fifth.

Guided practice

The rule is aₙ = 2 × 3^(n−1). Find a₆ and say why this is justified.

Hint: Use n − 1 = 5.

Check the guided answer

a₆ = 2 × 3⁵ = 486. The explicit rule, rather than a guess from early values, fixes that term.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A pattern seen four times is a proof of every later term.”

Correction: A finite list supports a conjecture; a rule or structural argument is needed to determine later terms.

Extend and reflect

Invent a rule matching n² for n = 1, 2, 3 but not n = 4.

Guidance: Add (n−1)(n−2)(n−3) to n²; the added term is zero for n = 1–3 but not n = 4.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Extend a stated rule. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Choose, test and use an appropriate sequence model

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Classify a sequence using differences and ratios
Calculate and interpret a future term with model limits

Vocabulary: model · ratio · difference · triangular · domain

Retrieve previous learning

What are the first four triangular numbers?

Check: 1, 3, 6, 10.

Learn the concept

A fictional arrangement adds rows of 1, 2, 3, ... seats. After n rows the total is Tₙ = n(n + 1)/2; after 12 rows there are 12 × 13/2 = 78 seats. The differences between successive totals increase by one, so a constant-ratio model would be inappropriate.

A different fictional process triples each day from 2 units, giving 2, 6, 18, 54 and aₙ = 2 × 3^(n−1). Its day-five value is 162. In a real setting, constraints such as capacity could invalidate unchecked future extrapolation. State whether the question asks for a mathematical pattern or a physical prediction.

Worked model

List 1, 3, 6, 10 and calculate differences 2, 3, 4.
Relate each difference to one extra row.
Use T₁₂ = 12 × 13/2 = 78.
Check the model applies to whole row counts n ≥ 1.

Guided practice

A display has 4, 12, 36, 108 items by stage. Find stage six under the stated repeated-tripling rule.

Hint: Stage four is 108; multiply twice more.

Check the guided answer

Stages five and six have 324 and 972 items respectively; equivalently a₆ = 4 × 3⁵ = 972.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Every non-linear sequence must be geometric.”

Correction: Square, cube and triangular sequences are non-linear without a constant ratio.

Extend and reflect

For the rule n² + n + 1, find stage 10 and compare its increase from stage 9.

Guidance: a₉ = 91 and a₁₀ = 111, so the increase is 20; first differences rise by 2 each stage.