Figurate, geometric and Fibonacci-type sequences · Foundation
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| recursive rule/rɪˈkɜːsɪv ruːl/ | 递推规则 | dì tuī guī zé |
Does the dot pattern add or multiply?
- A growing dot pattern adds a longer row each time. How can a diagram distinguish it from repeated multiplication?
- This lesson studies recursive rule 递推规则: A rule that generates new terms from preceding terms.
Choose the mathematical structure
- Record positions and terms separately. Triangular numbers add 1,2,3,...; square and cube numbers use n² and n³. A geometric sequence multiplies by a common positive ratio. A Fibonacci-type sequence starts with stated terms and then adds its two predecessors. A supplied recursive rule must include enough initial values.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines recursive rule?
A rule that generates new terms from preceding terms.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Triangular terms are 1,3,6,10,15; square terms 1,4,9,16,25; cube terms 1,8,27,64,125. The geometric sequence 2,6,18,54,... has ratio 3 and nth term 2×3^(n-1). Starting 2,3 and adding the previous two gives 2,3,5,8,13. For u next=2u+1 from u₁=1, the next terms are 3,7,15.
Figurate, geometric and Fibonacci-type sequences
Record positions and terms separately
Compare the model with the worked case and explain one change.
Find the fifth triangular number.
1+2+3+4+5=15.
Test a tempting shortcut
- A nonconstant first difference does not mean a pattern is random. A geometric ratio is not a common difference. Check the starting index when writing a position rule.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every sequence with increasing terms has a constant difference. This claim is false. Explain which definition or assumption it violates.
Find the next term after 2,6,18.
Multiply 18 by the common ratio 3.
Every sequence with increasing terms has a constant difference.
A nonconstant first difference does not mean a pattern is random. A geometric ratio is not a common difference. Check the starting index when writing a position rule.
Interpret a new situation
- AQA A23/A24 Foundation includes these patterns and positive rational geometric ratios. Higher also permits surd ratios and derives quadratic nth terms in the next lesson. Infinite-series sums are not part of this GCSE lesson.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the next term after 2,3,5,8 in a Fibonacci-type sequence.
Add the previous two: 5+8=13.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.2. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A rule that generates new terms from preceding terms. Choose the relationship, show the method, check its assumptions and interpret the result.