Chapter 2 · Sequences · Concept 3
2-13 Summation Symbol Σ (3)
Two more things go into the toolbox. One of them — the sum of cubes — is the prettiest fact in the chapter, because it turns out to be a square. The other is not new at all: it is the geometric sum from lesson 2-9, wearing a Σ.
Guiding question
What else can go inside a Σ, and where have you seen it before?
Learning outcome. Use the summation symbol to represent sums, and calculate their values.
The sum of cubes
Σ₁ⁿ k3 = {½n(n + 1)}2
which is just the square of Σk — remember it that way and you never have to learn it
And geometric sums, in Σ form
Σ₁ⁿ a rk−1 is the sum of the first n terms of a geometric sequence with first term a and ratio r. So it is lesson 2-9, unchanged:
r less than 1
a(1 − rⁿ)/(1 − r)
r more than 1
a(rⁿ − 1)/(r − 1)
The two shapes to recognise:
aₙ = rn−1 = 1 × rn−1 → first term 1, ratio r
aₙ = rn = r × rn−1 → first term r, ratio r
Every cube k³ is k copies of a k × k square. Lay them all out and they tile a perfect square — whose side is 1 + 2 + … + n.
the cubes added up
1³ + 2³ + 3³ + 4³ = 100
the square's side, squared
10² = 100
Σk³ = (Σk)² = {½n(n + 1)}²
A student who can do lesson 2-9 perfectly will still stall on Σ 4k — because it does not look like a sequence until you write it in the standard shape. Do that first, every time.
Σ 4ᵏ⁻¹
What are the first term and the common ratio here?
(a) Find Σ₁ⁿ k2(k + 1).
- 1
Multiply the bracket out. Nothing can split until you do.
Σ (k3 + k2)Same rule as last lesson — a product does not split, so remove it.
Work with a partner. For the geometric ones, both say what a and r are before either of you writes a formula. If you disagree, rewrite it as a rᵏ⁻¹ together.
Find the following sums.
Σ from r = 5 to 9 of (2r + 7)
Σ 4ᵏ and Σ 4ᵏ⁻¹ look almost the same, but their answers differ by a factor of 4. Where exactly does that factor come from, and could you get one answer from the other without redoing the work?
Write out the first three terms of each.
Reading Σ rᵏ as first term 1.
rᵏ = r × rᵏ⁻¹, so the first term is r. Rewrite it in the standard shape before you touch the formula.
Forgetting to square the whole of ½n(n+1).
{½n(n+1)}² is ¼n²(n+1)². Both the half and the bracket get squared.
Stopping before the second factorising.
3n² + 7n + 2 factorises into (n + 2)(3n + 1). The book asks for the fully factored form.
Splitting a product.
k²(k + 1) must be multiplied out into k³ + k² first. The rule from lesson 2-12 has not changed.
Sign slips with a negative ratio.
1 − (−3) is 4, not −2. Write the bracket in before simplifying it.
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