Arithmetic and finite geometric sequences · 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
- An arithmetic sequence adds a constant difference and has nth term a+(n-1)d. A geometric sequence multiplies by a constant ratio and has nth term ar^(n-1). Check the starting position and keep a surd ratio exact when one is supplied.
- 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 5,8,11,... the nth term is 3n+2, so u₈=26. For 2,6,18,... the common ratio is 3 and u_n=2×3^(n-1), giving u₄=54. The geometric sequence 1,√2,2,2√2,... has ratio √2. These are finite-term calculations; no infinite-series sum is used.
Arithmetic nth terms
u_n=5+3(n-1)
Use the arithmetic widget for the arithmetic case. Compare its constant difference with the geometric sequence in the worked example.
For a=5,d=3, find u_8.
The arithmetic eighth term is 5+7×3=26.
Test a tempting shortcut
- A geometric ratio is not a common difference. The exponent is n-1 when a is the first term at position 1. A finite pattern does not itself justify an infinite-sum formula.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A common ratio and a common difference describe the same operation. This claim is false. Explain which definition or assumption it violates.
Find the common ratio of 2,6,18.
For 2,6,18,... the ratio is 6/2=3.
A common ratio and a common difference describe the same operation.
A geometric ratio is not a common difference. The exponent is n-1 when a is the first term at position 1. A finite pattern does not itself justify an infinite-sum formula.
Interpret a new situation
- AQA A23–A25 Higher uses arithmetic rules, geometric progression patterns and quadratic nth terms. Arithmetic-series formulae and infinite-series sums are excluded from 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 fourth term of 2,6,18,... .
The fourth term is 2×3³=54.
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 · Higher · 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.
The constant multiplier between consecutive terms of a geometric sequence. Choose the relationship, show the method, check its assumptions and interpret the result.