Sequences, series and recurrence · Higher
| English | 中文 | Pinyin |
|---|---|---|
| common ratio/ˈkɒmən ˈreɪʃɪəʊ/ | 公比 | gōng bǐ |
Does the change add or multiply?
- A saving plan adds ¥30 more each week; a population model grows by 5% each year. Equal differences and equal ratios need different models.
- This lesson studies common ratio 公比: The constant multiplier between consecutive terms of a geometric sequence.
Choose the mathematical structure
- For an arithmetic progression, u_n=a+(n-1)d and S_n=n(2a+(n-1)d)/2. For a geometric progression, u_n=ar^(n-1) and S_n=a(1-r^n)/(1-r). An infinite geometric sum exists only if |r|<1.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines common ratio?
The constant multiplier between consecutive terms of a geometric sequence.
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.
For a=5,d=3,n=8, u_8=5+7×3=26 and S_8=8(10+21)/2=124. For a=12,r=1/2, S infinity=12/(1-1/2)=24. For u_(n+1)=2u_n+1 with u_1=1, the next terms are 3,7,15.
Sequences, series and recurrence
For an arithmetic progression, u_n=a+(n-1)d and S_n=n(2a+(n-1)d)/2
Compare the model with the worked case and explain one change.
For a=5,d=3, find u_8.
u₈=5+(8-1)×3=26.
Test a tempting shortcut
- The first term has index 1, so the exponent is n-1. A sequence is a list; a series is a sum. A geometric sequence can alternate in sign and still converge.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every geometric series has a finite sum to infinity. This claim is false. Explain which definition or assumption it violates.
Find the sum of the first 8 terms with a=5,d=3.
S₈=8[2×5+7×3]/2=124.
Every geometric series has a finite sum to infinity.
The first term has index 1, so the exponent is n-1. A sequence is a list; a series is a sum. A geometric sequence can alternate in sign and still converge.
Interpret a new situation
- Explain whether the context justifies additive or multiplicative change. In finance, distinguish a single deposit from a stream of deposits before choosing a sum formula.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the infinite sum with a=12,r=0.5.
Since |r|<1, S∞=12/(1-0.5)=24.
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
- 4MA1 · Higher · 3. 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.
The constant multiplier between consecutive terms of a geometric sequence. Choose the relationship, show the method, check its assumptions and interpret the result.