Sigma notation, indices and changed limits
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| summation index/sʌˈmeɪʃn ˈɪndeks/ | chỉ số tổng |
How many terms does it contain, and what changes if we rename or shift its index?
- A sum starts at the third term.
- This lesson studies summation index 求和指标: The variable that takes each integer value between a sum’s stated limits.
The method
- In a sum from k=a to b, substitute each integer a,a+1,…,b into the summand and add the results.
- There are b−a+1 terms when a≤b.
- The summation index is a dummy variable, but other parameters keep their values.
- Renaming an index preserves the sum only when its bound occurrences are renamed consistently.
How many terms are in the sum from k=3 to k=7 inclusive?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
An inclusive index range contains b-a+1 values. Subtracting the endpoints without adding 1 loses the first term - the sum 5+7+9+11+13 has five visible terms.
A worked case
The sum from k=3 to 7 of (2k−1) is 5+7+9+11+13=45, with five terms. It also equals the sum from k=1 to 7 minus the sum from k=1 to 2: 49−4=45. With j=k−2, the lower limit becomes 1, the upper becomes 5 and 2k−1 becomes 2j+3, again giving 45. By linearity, the original sum is 2 times the sum of k from 3 to 7, minus five copies of 1: 2×25−5=45.
Sigma notation, indices and changed limits
In a sum from k=a to b, substitute each integer a,a+1,…,b into the summand and add the results
Check whether a sequence or sum argument preserves all its defining information.
Evaluate the sum of (2k−1) from k=3 to 7.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Add 5+7+9+11+13=45.
Where it goes wrong
- An inclusive sum from 3 to 7 has five terms, not four or seven.
- Subtract the prefix through a−1, not through a, when changing the lower limit.
- A constant is added once per index value: the sum of c from a to b is c(b−a+1).
- Shifting the index changes both bounds and the summand.
Not true: Renaming or shifting an index never requires changing the summand. Shifting the index by k means replacing r by r+k in the summand as well.
How many terms occur in that sum?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Count inclusive integer indices: 7−3+1=5.
Renaming or shifting an index never requires changing the summand.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Shifting the index by k means replacing r by r+k in the summand as well.
Applying it
- For the sum of (2k−1) from k=1 to n, the index k changes while n is the fixed upper bound; the result is n² for positive integer n.
- Thus a tail from k=m to n is n²−(m−1)² for 1≤m≤n.
- Expand a few terms to check a symbolic manipulation before applying a sum formula.
After j=k−2, what is the new upper limit?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
When k=7, j=7−2=5.
Match each sigma manipulation to its correct form.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The index is a dummy variable: renaming or shifting it must change the bounds and the summand consistently.
In your course
- AQA A-level Mathematics (7357), specification section D.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Summation index: The variable that takes each integer value between a sum’s stated limits.
- In a sum from k=a to b, substitute each integer a,a+1,…,b into the summand and add the results.
- Watch for: an inclusive sum from 3 to 7 has five terms, not four or seven.