Arithmetic sequences and nth terms · Core
| English | 中文 | Pinyin |
|---|---|---|
| common difference/ˈkɒmən ˈdɪfrəns/ | 公差 | gōng chāi |
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 difference 公差: The constant added between consecutive terms of an arithmetic sequence.
Choose the mathematical structure
- Find a constant difference for an arithmetic sequence. Its nth term is a+(n-1)d. A term-to-term rule describes how to reach the next term; a position-to-term rule gives a term directly.
- 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 difference?
The constant added between consecutive terms of an arithmetic 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 5,8,11,14,... the common difference is 3. The nth term is 5+3(n-1)=3n+2. At n=8, u₈=26. To find the position of 62, solve 3n+2=62, giving n=20.
Arithmetic sequences and nth terms
Find a constant difference for an arithmetic sequence
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.
The first term of a sequence always has index zero. This claim is false. Explain which definition or assumption it violates.
At what position does 62 occur in u_n=3n+2?
Solve 3n+2=62: 3n=60, hence n=20.
The first term of a sequence always has index zero.
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
- Generate several terms and check a proposed nth-term rule. Infinite geometric series and advanced sum formulae are excluded from this Foundation/Core lesson.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find u_4 for u_n=3n+2.
u₄=3×4+2=14.
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
- 9260 · Core · 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 added between consecutive terms of an arithmetic sequence. Choose the relationship, show the method, check its assumptions and interpret the result.