Sums of Integers, Squares & Cubes

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- This is important – if the top and bottom limits don’t all match then the relation is no longer valid!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 $\sum_{r = 1}^{n} a = a + a + a + . . . (n times) = n \times a$

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, $\sum_{}^{} 2 r^{3} - 3 r^{2} - 8 r - 3 = 2 \sum_{}^{} r^{3} - 3 \sum_{}^{} r^{2} - {8 \sum}_{}^{} r - \sum_{}^{} 3$

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

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)