Quadratic nth terms and surd-ratio progressions · Higher
| English | 中文 | Pinyin |
|---|---|---|
| second difference/ˈsekənd ˈdɪfrəns/ | 二阶差分 | èr jiē chā fēn |
What does the second difference reveal?
- The terms 3,8,15,24 grow by 5,7,9. What rule predicts the next term without building every earlier term?
- This lesson studies second difference 二阶差分: The difference between consecutive first differences of a sequence.
Choose the mathematical structure
- A constant second difference signals a quadratic rule an²+bn+c. Its value is 2a. Subtract an² from the terms; fit the remaining linear rule and test at several positions. A geometric sequence with a surd ratio still multiplies by the same exact factor; keep root values exact.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines second difference?
The difference between consecutive first differences of a 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 3,8,15,24, first differences are 5,7,9 and second difference 2, so a=1. Subtract n² at n=1,2,3,4 to get 2,4,6,8=2n. Thus u_n=n²+2n and u₅=35. For 1,√2,2,2√2,... the ratio is √2 and u_n=(√2)^(n-1). Check u₃=2 rather than rounding the root repeatedly.
Quadratic nth terms and surd-ratio progressions
A constant second difference signals a quadratic rule an²+bn+c
Compare the model with the worked case and explain one change.
Find a when the second difference is 2.
2a=2, so a=1.
Test a tempting shortcut
- The second difference equals 2a, not a. A quadratic rule must be checked against the first terms; there can be a nonzero constant c. A finite pattern alone does not prove a unique rule without the stated sequence family.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The coefficient of n² always equals the second difference. This claim is false. Explain which definition or assumption it violates.
Find u₅ for u_n=n²+2n.
25+10=35.
The coefficient of n² always equals the second difference.
The second difference equals 2a, not a. A quadratic rule must be checked against the first terms; there can be a nonzero constant c. A finite pattern alone does not prove a unique rule without the stated sequence family.
Interpret a new situation
- AQA A25 Higher derives quadratic nth terms; A24 permits surd-ratio sequences. Foundation recognises and generates quadratic patterns without this general derivation.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the third term of the geometric sequence starting 1 with ratio √2.
1×√2×√2=2.
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 difference between consecutive first differences of a sequence. Choose the relationship, show the method, check its assumptions and interpret the result.