lesson
Recognise how a sequence is generated
1 · Learn
An arithmetic sequence has a constant first difference and a linear nth term. If the common difference is d, start with dn and adjust the constant to match a known term.
Triangular, square and cube numbers come from geometric or power patterns. A Fibonacci-type sequence adds earlier terms according to a stated rule. A geometric progression multiplies by a constant positive rational ratio rather than adding a constant.
A few matching terms do not uniquely prove a rule. State the rule and test it beyond the terms used to find it.
2 · Worked example
5,8,11,14 has difference 3. The rule 3n+2 gives 5 when n=1 and 14 when n=4. The next term is 17.
3 · Your turn
For 7,11,15,19, find the next two terms and the nth term. Name the family 1,4,9,16.
Check your answer
23,27; nth term 4n+3. The second sequence is the square numbers n².
4 · Apply your learning
Create one arithmetic and one geometric sequence with the same first term. Explain which test distinguishes them.
Adult guidance and safety
This is England national-curriculum core teaching, not an exam-board specification. Check prerequisite understanding, require clear working and use additional varied practice before judging fluency.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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