Chapter 2 · Sequences · Concept 3
2-12 Summation Symbol Σ (2)
Last lesson Σ was only a way of writing things down. Now it starts doing work. Three sums are worth knowing by heart, and one splitting rule lets you break any tangle into pieces that are each one of those three.
Guiding question
Which sums are worth learning, and how do you reach them?
Learning outcome. Use the summation symbol to represent sums, and calculate their values.
The three formulas
Σ₁ⁿ c
= nc
c is any plain number
Σ₁ⁿ k
= ½n(n + 1)
the counting numbers
Σ₁ⁿ k²
= ⅙n(n + 1)(2n + 1)
the square numbers
And the splitting rules
Σ p·aₖ = p · Σ aₖ — a plain number comes outside
Σ (aₖ + bₖ) = Σ aₖ + Σ bₖ — a sum splits into sums
Σ (aₖ − bₖ) = Σ aₖ − Σ bₖ — and so does a difference
The rule that does NOT exist:
Σ (aₖ × bₖ) ≠ Σ aₖ × Σ bₖ
Multiplying does not split. Check it with n = 2: Σk² is 1 + 4 = 5, but Σk × Σk is 3 × 3 = 9. That is why Σ k(5k − 2) has to be multiplied out first, before anything can be split.
Nothing on these pages is a new idea. Every question is the same journey — pull the Σ apart until each piece is one of the three, swap them in, and factorise at the end.
Σ (5k + 4)
the messy one you were handed
what the book wants
½n(5n + 13)
Take out the fraction and whatever the pieces share, then tidy only inside the bracket.
the same number, fewer marks
5n²/2 + 13n/2
Correct, but not finished — and much harder to check or use in the next part of a question.
How to spot what to take out. Look at the two pieces before you touch them: 5 × ⅙n(n+1)(2n+1) and 2 × ½n(n+1) both contain n(n + 1), and the worst fraction is ⅙. So take out ⅙n(n + 1) and see what is left.
(c) Find 1×3 + 2×8 + 3×13 + … + n(5n − 2).
- 1
Write it as a Σ first — you cannot use any rule until you have.
Σ from k = 1 to n of k(5k − 2)Check the pattern: k = 1 gives 1 × 3 ✓, k = 2 gives 2 × 8 ✓
Work with a partner. One of you does the splitting, the other does the factorising at the end. The middle bit is just looking things up.
Find the following sums.
Σ from r = 4 to 8 of (r2 + r + 5)
Why is there a rule for splitting Σ(aₖ + bₖ) but no rule for splitting Σ(aₖ × bₖ)? Test both on n = 2 with aₖ = bₖ = k.
Work out both sides of each claim as plain numbers.
Splitting a product.
Σ(aₖ × bₖ) is not Σaₖ × Σbₖ. Multiply the bracket out first, then split the sum that results.
Expanding at the end.
½n(5n + 13) is the answer. Multiplying it out into 5n²/2 + 13n/2 is the same number and loses marks.
Forgetting Σ of a plain number.
Σ₁ⁿ 4 is 4n, not 4. The k counts the repeats even when it never appears inside.
Using n when the top is a number.
For Σ₁²⁰, put 20 in wherever the formula has an n. Leaving an n in the answer means the question was not finished.
Starting the formula at the wrong place.
The three formulas all begin at k = 1. If a Σ starts at 4, work out the whole thing from 1 and subtract the first three.
Sign In to Keep Going
You've reached the end of the free preview. Sign in — and upgrade to Plus or Pro — to keep going with Mathematics.
Full Access
Unlock all 105 lessons