Sequences and nth terms

Your learning journey

Generate and recognise sequences. Find a linear nth-term rule and use it to decide whether a value belongs.

Work without a calculator except for an invited check. Keyboard - means the same as −. Attempt before opening a scaffold or intentional practice explanation.

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Ready.

Connect & Check

Term: number in a sequence. Position: index n, starting at 1 here. Arithmetic: constant difference. Geometric: constant multiplier. Fibonacci type: add the previous two terms.

Share

In pairs, predict the first step and compare your reasons before reading the worked solution: D1 Model: 5,8,11,... compares with 3,6,9,..., so nth term 3n+2. Find the rule for 7,10,13,... .

Present

For an arithmetic sequence, dn supplies the constant difference d. Compare the first term with d to find the adjustment. Thus 5,8,11,... is 3n+2. A term-to-term rule generates the next term; an nth-term rule finds any position directly.

Inspect differences and ratios before choosing a pattern. For linear sequences write dn+c, use n=1 to find c and test more terms. To test membership, solve for n and require a positive integer.

Apply

Optional check after A2: calculate 4×20−1=79 and (42+1)÷4=10.75. A non-integer answer does not become a sequence position by rounding.

Apply and extend

Access, Scaffold, Core and Stretch support the same goal. Stretch deepens reasoning for either qualification tier.

Recall & Review

Retrieval without notes

Six retrieval points. Enter a number, decimal, fraction such as 3/2, or mixed number such as 1 1/2. Equivalent numeric values are accepted. Unsupported notation goes to teacher review. For rounding, also communicate the requested accuracy in written work; numerical scoring cannot judge significant trailing zeros.

Not checked.

Independent exit evidence

P1–P2: five minutes, no notes, 4 marks. Your teacher reviews working and explanations. Full assessment solutions are not embedded.

Recall & Review and prepare

A checkpoint selects the next teaching step. Try without prompts, identify whether an error concerns units, operations, signs or structure, then practise a new question before rechecking after a delay.

Retrieve F1 again in one week. Bring your working and one question.

Higher alternative: Finding a quadratic nth term

Higher route: replace the 2-minute A3 feedback and the two 4-minute Challenge discussions with this 10-minute taught extension. Retain core P1–P2. Set HX1–HX2 as the final four minutes of that alternative, and use the shared challenges later if needed. The core route remains 60 minutes; the extension is not an additional compulsory ten minutes. Route choice reflects tier curriculum, not temporary support needs.

For 3, 6, 11, 18, 27 the first differences are 3, 5, 7, 9 and second differences are all 2. For a quadratic sequence an²+bn+c, the constant second difference is 2a. Thus a=1. Subtract n²: the remainders are 2, 2, 2, 2, 2. The rule is n²+2. Check n=1 and n=4: 3 and 18. This finds the quadratic rule under a quadratic-pattern assumption; a finite list alone cannot prove a unique continuation.

HX1–HX2: teacher review. Completion is not mastery.