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AS Level · Further Pure Mathematics 1 WFM01

Series

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The Big Idea: There are exact closed-form formulas for adding up the first n natural numbers, their squares, and their cubes — once you know all three, you can evaluate almost any polynomial series by splitting it term by term.

Chapter Overview

Key Concepts at a Glance

Sigma notation — compact shorthand for sums

1️⃣

Σr = ½n(n+1) — sum of natural numbers

2️⃣

Σr² = ⅙n(n+1)(2n+1) — sum of squares

3️⃣

Σr³ = ¼n²(n+1)² — sum of cubes

✂️

Writing non-1 lower limits as differences of two sums

🔗

Combinations — splitting complicated expressions term by term

The Three Standard Formulae — Quick Reference

Sum of Natural Numbers

Σ_r=1ⁿ r

= ½ · n(n + 1)

⚠ Not in Formula Book

Sum of Squares

Σ_r=1ⁿ r²

= ⅙ · n(n+1)(2n+1)

✓ Given in Formula Book

Sum of Cubes

Σ_r=1ⁿ r³

= ¼ · n²(n+1)²

✓ Given in Formula Book

Topic 1

Sigma Notation

What is Σ (sigma)?

Sigma notation is a shorthand for writing long sums. Think of it as a machine: you plug in values of r from the bottom number to the top number, calculate the expression each time, and add everything up.

General Form

Σ_{r = a}^b f(r) = f(a) + f(a+1) + f(a+2) + … + f(b)

r counts in whole numbers from the lower limit a up to the upper limit b

Example:

Σ_r=1⁵ (3r − 1) = (3·1−1) + (3·2−1) + (3·3−1) + (3·4−1) + (3·5−1)
= 2 + 5 + 8 + 11 + 14 = 40

Sigma Rules You Must Know

A constant just multiplies by n: Σ_r=1ⁿ k = kn (you're adding the same constant n times)
Coefficients of r come out of the sum: Σ(ar + b) = a·Σr + bn
Coefficients of r² come out: Σ(ar²) = a·Σr²
Coefficients of r³ come out: Σ(ar³) = a·Σr³
You can split sums: Σ(f + g) = Σf + Σg

🧠 Remember this pattern

Constants → multiply by n. Coefficients of r, r², r³ → pull out front. This is true for any combination of terms.

Dealing with a Non-1 Lower Limit

When the lower limit isn't 1 — for example Σ_r=5¹⁰ r — you can't directly use the standard formulas. The trick is to write it as a difference of two sums that both start at r = 1:

Lower Limit ≠ 1

Σ_r=a^b f(r) = Σ_r=1^b f(r) − Σ_r=1^a−1 f(r)

The upper limit of the second sum is one less than the original lower limit — be careful here!

Concrete example:

Σ_r=5¹⁰ r = (1+2+3+4+5+6+7+8+9+10) − (1+2+3+4)
= Σ_r=1¹⁰ r − Σ_r=1⁴ r
Note: upper limit of 2nd sum = 5 − 1 = 4, not 5!

⚠ Classic Mistake

Students often subtract up to the same lower limit. If your series starts at r = 5, the subtracted sum goes up to r = 4, not 5. The 1, 2, 3, 4 have to cancel — not the 5.

Practice Question 1

Write Σ_r=4¹² (2r + 1) as the difference of two sums that both start at r = 1. You do not need to evaluate them.

Topic 2

Sum of Natural Numbers

The Formula

The natural numbers are just the positive integers: 1, 2, 3, 4, … Adding the first n of them:

⚠ NOT in Formula Booklet — Memorise This

Σ_r=1ⁿ r = ½ · n(n + 1)

Multiply n by (n+1), then halve it. That's it.

💬 Why does this work?

Quick check: For n = 4: 1+2+3+4 = 10. Formula: ½ × 4 × 5 = 10. ✓

Topic 3

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Also in the full note

Worked Example — Using the Formula in a Proof
Worked Example — Non-1 Lower Limit
Worked Example — Even Cube Numbers
The Master Strategy
Worked Example — Full Combination Proof
Using a "Hence" Result

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