Finite sums and induction
| English | 中文 | Pinyin |
|---|---|---|
| sigma notation/ˈsɪɡmə nəʊˈteɪʃn/ | 求和符号 | qiú hé fú hào |
Can a polynomial replace an exponential nearby?
- Near a known point, a complicated function can be replaced by a polynomial approximation. The approximation needs a centre and a valid scale.
- This lesson studies sigma notation 求和符号: Notation that adds indexed terms over a stated finite range.
Choose the mathematical structure
- For k=1 to n, sum k=n(n+1)/2, sum k²=n(n+1)(2n+1)/6 and sum k³=[n(n+1)/2]². Differences between consecutive terms can telescope. In induction, verify the starting case and show the next case follows from the assumed case.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines sigma notation?
Notation that adds indexed terms over a stated finite range.
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 n=5, sum k=15, sum k²=1+4+9+16+25=55 and sum k³=1+8+27+64+125=225. In proving a sum formula, adding the (n+1)th term to the assumed expression must produce the formula with n replaced by n+1.
Finite sums and induction
For k=1 to n, sum k=n(n+1)/2, sum k²=n(n+1)(2n+1)/6 and sum k³=[n(n+1)/2]²
Compare the model with the worked case and explain one change.
Find the sum of k from 1 to 5.
1+2+3+4+5=15.
Test a tempting shortcut
- The sum of squares is not the square of the sum. State the index range and check the first case; an inductive step by itself is not a complete induction proof.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The sum of squared terms always equals the square of their sum. This claim is false. Explain which definition or assumption it violates.
Find the sum of k² from 1 to 5.
1²+2²+3²+4²+5²=55.
The sum of squared terms always equals the square of their sum.
The sum of squares is not the square of the sum. State the index range and check the first case; an inductive step by itself is not a complete induction proof.
Interpret a new situation
- FP1 finite series and induction support only. Maclaurin and Taylor approximation belong to later further-pure units and are excluded.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the sum of k³ from 1 to 5.
1³+2³+3³+4³+5³=225, also 15².
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
- edexcel IAL further mathematics; official unit FP1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- 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.
Notation that adds indexed terms over a stated finite range. Choose the relationship, show the method, check its assumptions and interpret the result.