Sums of Integers, Squares & Cubes

How can we use sigma notation?

When writing the sum of a series you can use sigma notation
- The series $u_{1} + u_{2} + u_{3} + . . . + u_{n - 1} + u_{n}$ can instead be written as $\sum_{r = 1}^{r = n} u_{r}$ or $\sum_{r = 1}^{n} u_{r}$
- This means “the sum of all the terms from $u_{1}$ to $u_{n}$ for the sequence described by $u_{r}$ "
- $\sum_{r = 1}^{3} 2 r + 3$ would mean $(2 (1) + 3) + (2 (2) + 3) + (2 (3) + 3) = 21$
Using the following relations, summations can be grouped together (or ungrouped) to make some calculations easier:

$\sum_{r = p}^{q} a f (r) + b = \sum_{r = p}^{q} a f (r) + \sum_{r = p}^{q} b = {a \sum}_{r = p}^{q} f (r) + \sum_{r = p}^{q} b$

$\sum_{r = p}^{q} a f (r) + b g (r) = \sum_{r = p}^{q} a f (r) + \sum_{r = p}^{q} b g (r) = {a \sum}_{r = p}^{q} f (r) + b \sum_{r = p}^{q} g (r)$

a and b are constants, and $f (r)$ and $g (r)$ are a functions of r
Note that the top and bottom summation limits ( $p$ and $q$ ) are the same for all the sums
- This is important – if the top and bottom limits don’t all match then the relation is no longer valid!
This can be very useful when $f (r)$ or $g (r) = r, r^{2}$ or $r^{3}$ , as $\sum_{}^{} b$ and $\sum_{}^{} r$ (or $r^{2}$ or $r^{3}$ ) are straightforward to find using formulae

A useful result to remember is that $\sum_{r = 1}^{n} a = a + a + a + . . . (n times) = n \times a$

What are the formulae for finding sums of integers, squares, and cubes?

There are several useful formulae for summing integers, square numbers, and cube numbers
The sum of the first $n$ natural numbers is given by
- $\sum_{r = 1}^{n} r = \frac{1}{2} n (n + 1)$
  - This formula is not given in the formula book
The sum of the first $n$ square numbers is given by
- $\sum_{r = 1}^{n} r^{2} = \frac{1}{6} n (n + 1) (2 n + 1)$
  - This formula is given in the formula book
The sum of the first $n$ cube numbers is given by
- $\sum_{r = 1}^{n} r^{3} = \frac{1}{4} n^{2} {(n + 1)}^{2}$
  - This formula is given in the formula book
  - Notice that this is equal to the formula for the sum of the first $n$ natural numbers, squared
Using the relations given above, a more complicated summation can often be broken down into sums of constants, natural numbers, squares, and cubes
- For example, $\sum_{}^{} 2 r^{3} - 3 r^{2} - 8 r - 3 = 2 \sum_{}^{} r^{3} - 3 \sum_{}^{} r^{2} - {8 \sum}_{}^{} r - \sum_{}^{} 3$

Examiner Tip

You can find summations using sigma notation on most advanced scientific calculators or graphics calculators – you can use this to check your answers
Bear in mind, however, that the question will normally require you to show your full working

Worked example

Find $\sum_{r = 1}^{7} (2 r + 1) (r - 3) (r + 1)$

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Sums of Integers, Squares & Cubes (Edexcel A Level Further Maths: Core Pure): Revision Note

How can we use sigma notation?

What are the formulae for finding sums of integers, squares, and cubes?

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1. Complex Numbers

1.1 Complex Numbers & Argand Diagrams

1.1.1 Introduction to Complex Numbers

1.1.2 Solving Equations with Complex Roots

1.1.3 Modulus & Argument

1.1.4 Modulus-Argument Form

1.1.5 Loci in Argand Diagrams

1.1.6 Regions in Argand Diagrams

1.2 Exponential Form & de Moivre's Theorem

1.2.1 Exponential Form

1.2.2 de Moivre's Theorem

1.2.3 Applications of de Moivre's Theorem

1.2.4 Roots of Complex Numbers

2. Matrices

2.1 Properties of Matrices

2.1.1 Introduction to Matrices

2.1.2 Determinants of Matrices

2.1.3 Inverses of Matrices

2.2 Transformations using Matrices

2.2.1 Transformations using a Matrix

2.2.2 Geometric Transformations with Matrices

2.2.3 Invariant Points & Lines

3. Further Algebra & Functions

3.1 Roots of Polynomials

3.1.1 Roots of Polynomials

3.1.2 Linear Transformations of Roots

3.2 Series

3.2.1 Sums of Integers, Squares & Cubes

3.2.2 Method of Differences

3.3 Maclaurin Series

3.3.1 Maclaurin Series

4. Hyperbolic Functions

4.1 Hyperbolic Functions

4.1.1 Hyperbolic Functions & Graphs

4.1.2 Logarithmic Forms of Inverse Hyperbolic Functions

4.1.3 Hyperbolic Identities & Equations

4.1.4 Differentiating & Integrating Hyperbolic Functions

5. Further Calculus

5.1 Volumes of Revolution

5.1.1 Volumes of Revolution

5.1.2 Modelling with Volumes of Revolution

5.2 Methods in Calculus

5.2.1 Improper Integrals

5.2.2 Mean Value of a Function

5.2.3 Integrating with Partial Fractions

5.2.4 Calculus involving Inverse Trig

5.2.5 Integration by Substitution

6. Further Vectors

6.1 Vector Lines

6.1.1 Equations of Lines in 3D

6.1.2 Pairs of Lines in 3D

6.1.3 Angle between Lines

6.1.4 Shortest Distances - Lines

6.2 Vector Planes

6.2.1 Equations of planes

6.2.2 Combinations of Lines & Planes

6.2.3 Combinations of Planes

6.2.4 Shortest Distances - Planes

7. Polar Coordinates

7.1 Polar Coordinates

7.1.1 Polar Coordinates

7.1.2 Calculus with Polar Coordinates

8. Differential Equations

8.1 First Order Differential Equations

8.1.1 Intro to Differential Equations

8.1.2 Solving First Order Differential Equations

8.1.3 Modelling using First Order Differential Equations

8.2 Second Order Differential Equations

8.2.1 Solving Second Order Differential Equations

8.2.2 Coupled First Order Linear Equations

8.3 Simple Harmonic Motion

8.3.1 Simple Harmonic Motion

8.3.2 Damped or Forced Harmonic Motion

9. Proof