Sums of series inquiry
The prompt
Mathematical inquiry processes: Interpret and verify; reason; generate examples; prove. Conceptual field of inquiry: Sigma notation; sum of a series; summation formulae.
The prompt is suitable for the A-level Further Maths course. It invites students to compare the sums of two series.
The sums of the series are equal and the prompt is true. The teacher might show this by making n = 5, for example, when both sides give 0 + 2 + 6 + 12 + 20 = 40. To show that explicitly, the following illustration is part of the PowerPoint in the resources section.
Students would then test other values of n.
As with all inquiries on the website, the prompt does not require discrete knowledge to be taught beforehand. Indeed, students are intrigued by the sigma notation and, after attempts to deduce its meaning, a teacher's explanation based on those deductions is more meaningful and powerful.
In the notice and wonder phase at the start of the inquiry, students often compare the two sides of the equation by saying what is the same and what is different:
"The symbol is the same as are the variables r and n."
"The r and terms in n are one less on the left-hand side."
"The algebraic expressions (general terms) are similar, but the (r + 1) is replaced by (r - 1) on the right-hand side."
Explanation
Why is the prompt true? The teacher should start an explanation by pointing out that the general terms are deceptive. The r on the left-hand side has become (r - 1) on the right-hand side and (r + 1) has changed to r - that is, both parts are one less on the right-hand side. In other words, f(r) is transformed to f(r - 1).
At the same time the series on the right-hand side starts and finishes at terms that are one more than the first and last term on the left-hand side. The two changes 'counteract' or cancel each other out. Such an explanation opens the way for students to create more general terms with their limits that sum to the same amount.
In a structured inquiry, the class could follow the lines of inquiry below; in a more open inquiry, students might select an approach from the regulatory cards.
November 2023
Lines of inquiry
1. Verify
Firstly, students verify that the sums of the series are equal, being 1(0) + 2(1) + 3(2) + 4(3) + 5(4) + 6(5) + 7(6) + 8(7) + 9(8) + 10(9) in the three cases. They can then extend the list for n = 10 or change the value of n.
Secondly, students find a series equivalent to those in the prompt by choosing between two options (see table below).
One option is correct and the other shows the misconception that if the limits increase or decrease by k, then f(r) becomes f(r + k) or f(r - k) respectively.
Both tasks reinforce the explanation about the changes to the general terms and limits.
2. Generate examples
Students aim to generate general terms with limits that are equivalent to the ones in the list.
3. Prove
To end the inquiry, students prove the prompt is true using summation formulae for the sum of n natural numbers and the sum of the squares of n natural numbers. They can also use the summation formulae for higher powers on their own general terms. See the mathematical notes for examples of proof.
