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

Identify an ordered term and its position
Test a proposed pattern against several terms

Vocabulary: sequence · term · position · pattern · difference

Retrieve previous learning

What does it mean for terms in a sequence to be ordered?

Check: Their positions are fixed, so the first number and each later number can be identified.

Learn the concept

A sequence is an ordered list of terms. Each term has a position: first, second, third and so on. In 4, 7, 10, 13, the third term is 10. The position is not the same as the value: the third term need not equal 3. Write the positions underneath a list before describing a pattern; this keeps the rule tied to the correct terms.

The differences between consecutive terms can reveal a pattern. For 4, 7, 10, 13 each difference is +3, so adding three would give 16 next. A different list, 2, 4, 8, 16, doubles each time; its differences are 2, 4 and 8, so calling it a constant-difference sequence would be wrong. Calculate more than one change before naming a rule.

A rule should explain the supplied terms, not merely one convenient transition. Two examples can share an early start and then diverge. For instance 2, 4, 6, 8 uses +2, while 2, 4, 8, 16 uses ×2 after the first pair. With only a few terms, several invented rules may fit; state the simplest supported rule and the evidence used.

A missing term can be checked by restoring the pattern on both sides. In 5, 9, __, 17, equal differences of four give 13, and 13 to 17 confirms it. If a problem does not state the rule and the pattern is ambiguous, say what assumption you make. This is an algebra habit: distinguish a justified prediction from a guaranteed unique continuation.

Worked model

Label positions 1 to 4 under 4, 7, 10, 13.
Subtract neighbouring terms to find +3 three times.
Predict the fifth term as 13 + 3 = 16.
Check the new difference is still +3 and state the rule assumption.

Guided practice

What is the missing term in 6, 10, __, 18 if the difference is constant?

Hint: Check both gaps.

Check the guided answer

The common difference is +4, so the third term is 14; 14 + 4 = 18 confirms the rule.

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 third term must have value three.”

Correction: Third names the position; its value depends on the sequence rule.

Extend and reflect

Give two different rules that start 1, 2, 3 but disagree at the fourth term.

Guidance: For example use +1 throughout, then invent a rule that changes after position three; explain why more information is needed.

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

This lesson practises: Identify an ordered term and its position. 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 8 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Generalise a growing pattern

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

Move from an example to a verbal general rule
Check a generalisation against new positions

Vocabulary: generalise · rule · constant · increase · counterexample

Retrieve previous learning

What evidence shows 6, 10, 14, 18 follows a simple repeated-addition pattern?

Check: The differences between adjacent terms are each four, so adding four predicts the next entry under that assumption.

Learn the concept

Generalising means describing what stays true beyond one or two examples. In a staircase made from 3, 6, 9 and 12 counters, each new stage adds three counters. The verbal rule 'start at three and add three for each new stage' predicts stage five as 15. A diagram can help count, but the number of counters must be tied to the stage number.

A term-to-term description starts from a known term and tells how to get the next. It is useful for nearby terms, but reaching stage 100 by repeated addition is slow. A position-based statement connects a term directly to its stage. For the staircase above, stage n has three times n counters: 3n. Test n = 1 and n = 4 before trusting the expression.

Not every pattern grows by a fixed difference. A square array with side lengths 1, 2, 3 and 4 contains 1, 4, 9 and 16 counters. Its successive differences are 3, 5 and 7, so 'add the same amount' fails. The area model suggests n × n = n² at stage n. A visual justification should explain why this rule fits every square stage, rather than guess from the first four values.

A counterexample can test a claimed general rule. If somebody says every increasing sequence has a constant difference, the square numbers disprove it because their gaps change. A few numerical matches do not prove a rule beyond the supplied range, but a defined construction can support a general statement. Name the construction or stated rule whenever a task asks for a later term.

Worked model

Count 3, 6, 9, 12 counters at stages 1 to 4.
Describe the repeated increase of three counters per stage.
Connect stage number to total: stage n has 3n counters.
Test n = 4 gives 12 and predict stage 7 gives 21.

Guided practice

A square array has side length 5. How many counters, and why?

Hint: Use area, not a constant difference.

Check the guided answer

It has 5 × 5 = 25 counters. The construction defines the fifth stage as a five-by-five square, so the rule is n².

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

“An increasing pattern must add the same amount every time.”

Correction: Square numbers increase while their differences change; check the rule or construction.

Extend and reflect

Explain why 3n gives every stage of a three-counter-per-stage staircase.

Guidance: Partition stage n into n groups of three, then test small and large stage numbers.

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

This lesson practises: Move from an example to a verbal general 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 8 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Use terms, positions and a defined 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

Use sequence language precisely
Generate terms from a stated position rule

Vocabulary: term · index · nth term · finite · infinite · rule

Retrieve previous learning

What does n represent in the expression 3n for a growing stage?

Check: n is the position or stage number, so 3n gives the number of counters at stage n.

Learn the concept

A sequence is an ordered succession of terms described by a rule. Positions are usually counted from 1 in the school examples here: a₁ is the first term, a₂ the second and aₙ the term at position n. The notation aₙ names a value; it does not mean the value must equal n. If aₙ = n + 4, the first four terms are 5, 6, 7 and 8.

A finite sequence has a stated last position, such as the first five multiples of four: 4, 8, 12, 16, 20. An infinite sequence has no last position specified and continues under its rule. Ellipsis can indicate continuation, but the dots alone do not tell a unique rule. A definition like aₙ = 4n for positive integer n makes the intended continuation explicit.

A position-to-term rule gives any term directly once n is known. By contrast, a term-to-term rule gives the next term from an earlier one and requires a starting value. The instruction 'add three' alone could generate many sequences; start at 2 gives 2, 5, 8, whereas start at 7 gives 7, 10, 13. Record both a start and repeated step when using a recursive description.

Check a rule by substituting several positive integer positions. For aₙ = 2n − 1, positions 1, 2, 3 and 4 give 1, 3, 5 and 7. Substitution at n = 0 may be algebraically possible but is not a listed term if the domain starts at n = 1. In a question about an nth term, state the allowed positions before claiming membership.

Worked model

Write n = 1, 2, 3 and 4 as the allowed first positions.
Substitute each n into aₙ = 2n − 1.
Obtain 1, 3, 5 and 7 in that order.
Check n = 1 gives the stated first term and explain why n = 0 is outside this example's domain.

Guided practice

Give the first three terms for aₙ = n + 4, with n starting at 1.

Hint: Substitute 1, 2 and 3.

Check the guided answer

The first three terms are 5, 6 and 7, because 1 + 4 = 5, 2 + 4 = 6 and 3 + 4 = 7.

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

“If aₙ = 2n − 1, n = 0 must be its first term.”

Correction: These school sequences usually specify positive integer positions; a₁ is first unless another domain is defined.

Extend and reflect

Compare aₙ = 4n for n = 1 to 5 with aₙ = 4n for every positive integer n.

Guidance: The formula is the same but one domain defines five terms and the other continues indefinitely.

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

This lesson practises: Use sequence language precisely. 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 8 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Generate terms from a repeated step

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

Apply a starting term and recurrence accurately
Recognise when a recurrence does not have a constant difference

Vocabulary: term-to-term rule · recurrence · start · previous term · operation

Retrieve previous learning

Why does 'add three' not identify one unique sequence?

Check: Without a starting term there are many possible lists that add three between neighbours.

Learn the concept

A term-to-term rule begins with a specified first term and gives an instruction for obtaining the next term from the current one. Starting at 5 and adding 4 each time gives 5, 9, 13, 17 and 21. Label each output before repeating the instruction: adding 4 to the previous term is different from adding 4 to the first term every time.

A multiplicative recurrence behaves differently. Starting at 64 and dividing by 2 each step gives 64, 32, 16, 8 and 4. The differences are −32, −16, −8 and −4, so a constant-difference claim is false even though the values steadily fall. State whether the rule says add, subtract, multiply or divide, and apply its order consistently.

Some recurrences combine operations. If the rule is 'multiply the previous term by 2, then add 1', starting at 1 gives 1, 3, 7, 15. Reversing the order to 'add 1 then multiply by 2' produces 1, 4, 10, 22 instead. Brackets make the instruction clear: next = 2 × current + 1 is not the same as 2 × (current + 1).

Repeated application is efficient for a few next terms, but to reach term 100 it may require many steps. A direct position-to-term rule can be preferable when one is available. Do not assume a simple formula merely because the first four terms look familiar. A complete description includes the initial term, the recurrence and the allowed positions.

Worked model

Record first term 5 at position 1.
Apply 'add four' to obtain position 2 as 9.
Repeat from each new term to obtain 13, 17 and 21.
Check all adjacent differences are four and state that 5 is indispensable to the rule.

Guided practice

Start at 64 and halve the previous term three times. What are the first four terms?

Hint: Each result becomes the next input.

Check the guided answer

The terms are 64, 32, 16 and 8; each term after the first is half its immediate predecessor.

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 term-to-term rule works without any starting value.”

Correction: The repeated operation needs an initial term to determine a specific sequence.

Extend and reflect

Can the rule 'halve the previous term' continue indefinitely from 64?

Guidance: In rational numbers it can continue with fractions; the answer depends on the number domain and whether only whole terms are allowed.

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

This lesson practises: Apply a starting term and recurrence 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 8 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Generate a term directly from its 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

Substitute a position into an nth-term rule
Compare direct and repeated-step descriptions

Vocabulary: position-to-term · nth term · substitute · expression · positive integer

Retrieve previous learning

What extra information must accompany the term-to-term rule 'add five'?

Check: A starting term is required; then each new term is five more than the previous one.

Learn the concept

A position-to-term rule uses a term number n as its input. For aₙ = 2n + 3 with n = 1, 2, 3, ... the terms start 5, 7, 9 and 11. Compute each by substituting its own position: the tenth term is 2 × 10 + 3 = 23. The expression is a direct formula, not an instruction to add 2n + 3 to the previous term.

The order of operations matters. For aₙ = 3n − 2, substitute n = 5 to get 3 × 5 − 2 = 13. Writing 3 × (5 − 2) would give 9 and represents a different rule. Use a small table with headings position n and term aₙ to make the link visible and to catch a rule that fails the given first term.

A direct rule may also be nonlinear. For aₙ = n², terms 1 through 4 are 1, 4, 9 and 16; position 10 gives 100 without finding positions 5 through 9. This demonstrates the advantage of direct substitution for a distant term. It does not make every sequence arithmetic: here differences grow rather than stay fixed.

The allowed values of n are part of the rule. In this course positions begin at positive integers unless a question states otherwise. If a formula produces a value at n = 0 or n = 1.5, that calculation may be algebraically valid but is not necessarily a term. When using a formula in a context such as rows or stages, connect n back to a meaningful whole-number position.

Worked model

Write n = 1, 2 and 3 in a position column.
Substitute each into aₙ = 2n + 3 to obtain 5, 7 and 9.
Substitute n = 10 directly to obtain 23.
Check the first three terms differ by two without confusing this with the direct formula.

Guided practice

Find term 5 for aₙ = 3n − 2.

Hint: Multiply before subtracting.

Check the guided answer

a₅ = 3 × 5 − 2 = 13, using the fifth position as the input.

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 expression tells me what to add to the previous term.”

Correction: It gives the term value directly from position n; a recurrence is a different kind of rule.

Extend and reflect

Describe a term-to-term rule for aₙ = 2n + 3.

Guidance: Find the first term and common difference; state both start at five and add two.

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

This lesson practises: Substitute a position into an nth-term 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 8 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Recognise a fixed common difference

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 arithmetic sequences from their differences
Continue increasing, decreasing and constant arithmetic lists

Vocabulary: arithmetic sequence · common difference · constant sequence · increasing · decreasing

Retrieve previous learning

How do the differences in 1, 4, 9, 16 show this is not arithmetic?

Check: They are 3, 5 and 7, so no single common difference links every adjacent pair.

Learn the concept

An arithmetic sequence has the same difference between every pair of neighbouring terms. In 7, 11, 15, 19 the common difference is +4. The word arithmetic describes the repeated addition structure, including subtraction as addition of a negative value. It does not mean every term must be positive or the sequence must increase.

A decreasing example is 17, 13, 9, 5 with common difference −4. Checking only the first gap is insufficient; calculate several differences. A list such as 3, 6, 12 has differences 3 and 6 and is not arithmetic, even though its first change was a neat whole number. If a list has one missing value, use both neighbouring gaps to test it.

A constant sequence such as 8, 8, 8, 8 is arithmetic with common difference zero. A positive difference gives increasing terms and a negative difference gives decreasing terms. These descriptions concern the ordered list, not whether an individual value is negative: −5, −2, 1, 4 increases even though its first entries are below zero.

Once the common difference is known, extend by repeated addition. To find a remote position efficiently, recognise that from the first term to term n there are n − 1 jumps. For 2, 6, 10, ... term 6 is 2 plus five jumps of four, or 22. This prepares the direct arithmetic formula in the next lesson; it also prevents the common off-by-one error of counting six jumps to reach position six.

Worked model

Subtract 11 − 7, 15 − 11 and 19 − 15.
Verify all three differences equal +4.
Label the sequence arithmetic and increasing.
Extend with 23, then check 23 − 19 = 4.

Guided practice

Is 17, 13, 9, 5 arithmetic? Give the common difference.

Hint: Check each adjacent subtraction.

Check the guided answer

Yes. Each new term is four less than the previous one, so the common difference is −4.

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

“An arithmetic sequence must increase by a positive whole number.”

Correction: Its common difference may be positive, negative or zero and terms can be fractional.

Extend and reflect

Give an arithmetic sequence of five fractions with common difference one half.

Guidance: Start at a chosen fraction and add 1/2 each time, checking all four gaps.

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

This lesson practises: Classify arithmetic sequences from their differences. 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 8 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Express an arithmetic term from its 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

Derive aₙ = a₁ + (n − 1)d
Use the common difference to simplify a direct formula

Vocabulary: first term · common difference · nth term · equivalent expression · linear

Retrieve previous learning

How many common-difference jumps are there from term 1 to term 6?

Check: There are five jumps, one fewer than the destination position number.

Learn the concept

An arithmetic sequence starts with a first term a₁ and changes by a fixed difference d. To reach term n from term 1 requires n − 1 changes, so aₙ = a₁ + (n − 1)d. For a first term of 7 and difference 4, a₈ = 7 + 7 × 4 = 35. The factor n − 1 counts the gaps, not the number of listed terms.

Expand the same rule if useful: 7 + 4(n − 1) = 7 + 4n − 4 = 4n + 3. Both forms produce the same value for every positive integer n. Check n = 1: the first form gives 7 + 0 and the second gives 4 + 3, both seven. The simplified form is useful for large positions and for testing membership.

A negative difference works without a new formula. Starting at 12 and subtracting 3 each time gives aₙ = 12 − 3(n − 1) = 15 − 3n. At n = 5 the value is 0. It is not correct to write 12 − 3n, which would give 9 at position one and lose the supplied first term.

A formula derived from a pattern is justified only under the assumed arithmetic rule. If later terms break the common difference, the formula no longer describes the actual list. Use a small table of position and term to check at least two supplied values, then state the domain of positive integer positions. An expression can also be checked algebraically by expanding brackets carefully.

Worked model

Record a₁ = 7 and d = 4 for 7, 11, 15, ... .
Write aₙ = 7 + (n − 1) × 4 because there are n − 1 gaps.
Expand to 4n + 3 without changing the rule.
Check n = 1 gives 7 and n = 8 gives 35.

Guided practice

Write a formula for 12, 9, 6, ... and find term 5.

Hint: The difference is negative three.

Check the guided answer

aₙ = 12 − 3(n − 1) = 15 − 3n, so a₅ = 15 − 15 = 0.

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

“Term n is the first term plus n common differences.”

Correction: From term one to term n there are n − 1 gaps, so use a₁ + (n − 1)d.

Extend and reflect

Give a₁ and d for aₙ = −5n + 18.

Guidance: Substitute n = 1 for a₁ and compare consecutive values to identify d.

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

This lesson practises: Derive aₙ = a₁ + (n − 1)d. 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 8 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Calculate distant arithmetic terms efficiently

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

Evaluate an nth-term rule at a large position
Check results using first term and common difference

Vocabulary: substitution · term position · common difference · negative term · check

Retrieve previous learning

What direct rule describes an arithmetic sequence beginning at 8 with difference 6?

Check: aₙ = 8 + 6(n − 1), equivalently aₙ = 6n + 2 for positive integer n.

Learn the concept

A direct nth-term rule avoids listing every previous term. For aₙ = 5n − 2, term 100 is 5 × 100 − 2 = 498. This is a value at position 100, not the one-hundredth multiple of five: the minus two is part of the rule. State the position before substituting so that a request for 'the hundredth term' is not confused with a term having value 100.

The first-term form can be equally efficient. A sequence begins at 11 with difference −2, so aₙ = 11 − 2(n − 1). At term 40 there have been 39 decreases: 11 − 78 = −67. Negative answers can be valid. A calculator sign error is caught by checking that terms near the fortieth position should be well below the initial eleven.

When the difference is fractional, keep it exact as a fraction if possible. A sequence beginning at 1/2 with difference 3/2 has aₙ = 1/2 + (n − 1) × 3/2. At n = 5 this is 1/2 + 6 = 13/2. Rewriting fractions as decimals is possible, but premature rounding can conceal exact structure.

Check a distant calculation by estimating the scale and by using an equivalent expression. If aₙ = 5n − 2, term 100 should be close to 500. A result of 50 or 5,002 suggests a substitution or place-value mistake. For a first-term formula, test n = 1 and one nearby term before trusting a distant output. This numerical check does not replace the algebraic justification.

Worked model

Identify n = 40, a₁ = 11 and d = −2.
Count 39 changes from term one to term forty.
Calculate 11 + 39 × (−2) = 11 − 78 = −67.
Check the result is below eleven and the previous term is −65.

Guided practice

Calculate term 25 for a sequence starting at 5 with difference 7.

Hint: There are 24 jumps.

Check the guided answer

a₂₅ = 5 + 24 × 7 = 173. Substitution into the equivalent rule 7n − 2 gives the same value.

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 negative answer means the nth-term calculation has gone wrong.”

Correction: A decreasing arithmetic sequence can pass through zero and have valid negative later terms.

Extend and reflect

For aₙ = 5n − 2, which position has value 498?

Guidance: Solve 5n − 2 = 498 and check that n is a positive integer.

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

This lesson practises: Evaluate an nth-term rule at a large position. 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 8 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Find an arithmetic nth-term rule from values

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

Use common difference as the coefficient of n
Find the adjustment that makes the first term match

Vocabulary: coefficient · constant · nth term · common difference · substitution

Retrieve previous learning

Why does 7, 11, 15 have formula 4n + 3 rather than 4n + 7?

Check: Its difference gives coefficient four, and n = 1 must give seven, so the adjustment is three.

Learn the concept

For an arithmetic sequence, the common difference is the coefficient of n in a linear rule aₙ = dn + c. In 4, 9, 14, 19 the difference is 5, so begin with 5n. At n = 1, 5n gives 5 but the actual first term is 4, requiring an adjustment of −1. Thus aₙ = 5n − 1. Test n = 2 to obtain 9.

A general shortcut is c = first term − difference, because a₁ = d + c. For 6, 10, 14, d = 4 and c = 6 − 4 = 2, so aₙ = 4n + 2. This is the expanded form of 6 + (n − 1) × 4. The shortcut follows from the first-term equation; it is not a guess about a 'zeroth term'.

The method works for decreasing and negative-valued lists. In 20, 17, 14, 11, the difference is −3 and c = 20 − (−3) = 23, giving aₙ = −3n + 23. At n = 3 the rule gives 14. A positive constant does not imply the terms increase; the coefficient of n controls the common change.

A short list might match a linear formula initially yet later depart from it, so the question must state or imply an arithmetic continuation. A valid answer names the difference, derives the adjustment and checks at least two supplied positions. The same procedure cannot be applied to 1, 4, 9, 16, because its changing gaps make a single coefficient d inadequate.

Worked model

Calculate differences in 4, 9, 14, 19 and obtain d = 5.
Start with 5n and compare its first value five with the actual four.
Subtract one to form aₙ = 5n − 1.
Check n = 2 gives nine and n = 4 gives nineteen.

Guided practice

Find the nth-term rule for 6, 10, 14, ... assuming an arithmetic continuation.

Hint: Start with four n; make position one equal six.

Check the guided answer

The difference is four and the adjustment is two, so aₙ = 4n + 2. It gives six at n = 1 and ten at n = 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

“The first listed term is always the constant c in dn + c.”

Correction: At n = 1 the formula gives d + c, so c = first term − d.

Extend and reflect

Can two different arithmetic rules share the same first two terms?

Guidance: The first two terms fix the common difference and first term, so under the arithmetic assumption the rule is fixed.

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

This lesson practises: Use common difference as the coefficient of n. 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 8 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Plot a sequence as discrete coordinate points

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 against term value on named axes
Use graph structure to describe an arithmetic sequence

Vocabulary: coordinate · horizontal axis · vertical axis · discrete · gradient

Retrieve previous learning

Find the first three terms of aₙ = 2n + 1.

Check: At positions one, two and three the values are three, five and seven.

Learn the concept

A sequence can be represented by plotting position n on the horizontal axis and term value aₙ on the vertical axis. For aₙ = 2n + 1, the first points are (1, 3), (2, 5), (3, 7) and (4, 9). Label axes and choose a scale that includes all supplied values, including negative values if the sequence decreases below zero. Reversing the axes changes what a plotted point means.

Arithmetic-sequence points lie on a straight-line pattern because each unit increase in n changes the term by a fixed amount. For aₙ = 3n + 1, the points (1, 4), (2, 7), (3, 10) rise three units for each one step to the right. The change is the common difference. The line through these points can help visualise the rule, but the sequence itself contains only the plotted positive-integer positions.

Joining the points without qualification can suggest values at non-integer positions such as n = 2.5. Those values may belong to a related continuous function, but they are not terms when the sequence domain is positive integers. The vertical intercept of the extended line can help identify c in dn + c, yet n = 0 is not automatically the first term. Distinguish a helpful graph extension from the actual list.

A decreasing arithmetic sequence has points that slope down as position increases. For 9, 7, 5, 3 the common difference is −2, so aₙ = −2n + 11 and the plotted points drop by two each step. Check a proposed coordinate by substituting its integer n into the rule. A point that merely looks close on a rough drawing is weaker evidence than an exact substitution.

Worked model

Label the horizontal axis position n and vertical axis term value aₙ.
Calculate a₁ = 4, a₂ = 7 and a₃ = 10 for aₙ = 3n + 1.
Plot (1, 4), (2, 7) and (3, 10) with equal axis intervals.
Explain that each one-step increase in n raises the value by three and only integer positions are sequence terms.

Guided practice

Does (4, 9) belong to the graph of sequence aₙ = 2n + 1?

Hint: Substitute n = 4.

Check the guided answer

Yes. At n = 4 the rule gives 2 × 4 + 1 = 9, and four is an allowed positive integer position.

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 point on a line drawn through the term points is a term of the sequence.”

Correction: The given sequence uses positive integer positions; a continuous line is an extension of the plotted pattern.

Extend and reflect

What is the vertical intercept of the line extension for aₙ = 3n + 1, and is it a first term?

Guidance: It is one at n = 0, but the first sequence term is four at n = 1.

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

This lesson practises: Plot position against term value on named axes. 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 8 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Prove whether a number is a sequence term

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

Solve aₙ = target for a candidate position
Check the position is in the sequence domain

Vocabulary: term membership · candidate position · positive integer · equation · verify

Retrieve previous learning

What positions make valid terms for the sequences in this unit?

Check: Positive integer positions 1, 2, 3 and so on unless a question explicitly defines a different domain.

Learn the concept

To test whether a value belongs to a sequence, set its nth-term expression equal to the target and solve for n. For aₙ = 4n + 1, test 29 by solving 4n + 1 = 29. Subtract one and divide by four to get n = 7. Since seven is a positive integer, 29 is term seven. Substitute again: 4 × 7 + 1 = 29.

An algebraic solution may fail the domain check. In aₙ = 4n + 5, asking whether 30 occurs gives 4n = 25 and n = 25/4 = 6.25. The expression has a real-valued solution, but a school sequence has only positive integer positions; there is no term 6.25. Therefore 30 is not a term. Do not round n to 6 or 7 and claim membership.

A negative common difference can still produce a valid target later. For aₙ = 18 − 2n, solve 18 − 2n = 2 to get −2n = −16, so n = 8. The eighth term is two. By contrast, a target that gives n = 0 is outside a sequence starting at one even if an extended line has that intercept.

Show enough working to justify both the algebra and the domain decision. An unsupported 'yes' can hide an error in the rearrangement, while a candidate integer without substitution can hide a sign error. A list of early terms may help estimate where to look, but a direct equation is more convincing for distant values and avoids stopping too soon.

Worked model

Set 4n + 1 = 29 for the proposed target.
Subtract one to obtain 4n = 28.
Divide by four to obtain n = 7, a positive integer.
Substitute back to verify 4 × 7 + 1 = 29.

Guided practice

Is 30 a term of aₙ = 4n + 5 for positive integer n?

Hint: Solve for n and check the domain.

Check the guided answer

No. The equation gives n = 25/4 = 6.25, which is not a positive integer position; do not round it.

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

“If solving for n gives a decimal, I can round to the nearest term.”

Correction: Sequence positions are discrete integers, so a fractional solution does not establish term membership.

Extend and reflect

For aₙ = 4n + 1, describe the values that can be terms.

Guidance: They are positive integers one more than a positive multiple of four; check the first term and domain.

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

This lesson practises: Solve aₙ = target for a candidate position. 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 8 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Model a context with an arithmetic 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

Define the counted position and write a valid sequence rule
Use the rule to solve a context question and explain limitations

Vocabulary: model · row · weekly amount · cumulative total · intersection

Retrieve previous learning

What evidence shows that 46 is term seven of aₙ = 7n − 3?

Check: Solving 7n − 3 = 46 gives integer n = 7, and substitution checks the value.

Learn the concept

A context must define what each term counts. Suppose a hall has 8 seats in row one and each later row has four more seats than the row before. The seats in row n are 8 + 4(n − 1) = 4n + 4, so row 15 has 64 seats. The formula counts seats in one row, not the total seats across the first fifteen rows. Label the quantity and unit before using a rule.

To find a row with 40 seats, solve 4n + 4 = 40. Then 4n = 36 and n = 9, an allowed whole row number. Check row one has eight and row nine has 40. A target that produces n = 9.5 would not describe a whole row in this model; it could be between two row sizes but is not an actual row.

Two arithmetic plans can be compared by making their positions comparable. If plan A gives 4n + 2 points and plan B gives 3n + 8 points in stage n, equality requires 4n + 2 = 3n + 8. Solving gives n = 6, and both yield 26 points. Before and after stage six, compare the expressions rather than assume the plan with the higher first value stays ahead.

A weekly contribution is different from a running balance. If a savings plan deposits £5 in week one and £3 more than the previous week's deposit each later week, the week-n deposit is 5 + 3(n − 1). The total saved by week n is the sum of deposits, not that one term. A mathematical model depends on the stated fixed increase; if real rows or contributions change, revise the rule rather than claiming predictions are guaranteed.

Worked model

Define n as the positive integer row number and identify eight seats in row one.
Use difference four to write seats in row n as 8 + 4(n − 1) = 4n + 4.
Substitute n = 15 to obtain 64 seats in row fifteen.
Explain that 64 is one row's size, not the cumulative hall total.

Guided practice

In the row model, which row has 40 seats?

Hint: Solve 4n + 4 = 40.

Check the guided answer

Subtract four and divide by four to obtain n = 9. Row nine has 4 × 9 + 4 = 40 seats.

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 of weekly deposits is the total saved by week n.”

Correction: It is one week's deposit; the total is the sum of all deposits up to that week.

Extend and reflect

Calculate the total seats in the first three hall rows and compare it with the third-row term.

Guidance: Add 8 + 12 + 16 = 36, while row three alone has 16; label both quantities.