# Summation properties

Learn about the main Summation properties and learn how to use it to solve various types of algebraic operations.

11/09/2023

## Introduction

The properties of the summation allow us to perform powerful algebraic simplifications on equations involving this type of mathematical symbology. This article divides the properties of the summation into two categories:

- first we present the properties of the summation applied to the elements of numeric sets.
- then, the properties applied to the polynomial functions are presented.

## Summation properties in numeric set elements

There are some interesting properties of summation for numeric set. These properties are very useful and allow us to save time and effort in solving various types of problems. Let’s go to the properties.

### Scalar product

Let X be a set of real values and α be any scalar (value used to multiply). The property states that:

The summation of the scalar multiplied by the term

is equal to scalar multiplied by the summation of the term.

Let’s go to the demonstration.

Considering the set **X = { 10 _{1}, 3_{2}, 5_{3}, 7_{4}, 2_{5}, 9_{6}, 4_{7}}**

(2.x_{3}) + (2.x_{4}) + (2.x_{5})

it’s the same as

2(x_{3} + x_{4} + x_{5})

(2.5) + (2.7) + (2.2)

it’s the same as

2(5 + 7 + 2)

10 + 14 + 4

it’s the same as

2.14

28

it’s the same as

28

A little exercise for you to practice.

Considering the set **Y = { 5 _{1}, 7_{2}, 9_{3}, 4_{4}, 10_{5}}**, show that:

### Addition and subtraction

Let X and Y be two distinct sets of real values. The property states that:

The summation of term X ± term Y

is equal the summation of the term X ± the sum of the term Y.

Let’s demonstrate (first with addition).

Considering the set **X = { 10 _{1}, 3_{2}, 5_{3}, 7_{4}, 2_{5}, 9_{6}, 4_{7}}** and

**Y = { 5**

_{1}, 6_{2}, 1_{3}, 4_{4}, 8_{5}, 3_{6}, 10_{7}}(x_{4}+y_{4}) + (x_{5}+y_{5}) + (x_{6}+y_{6})

it’s the same as

(x_{4} + x_{5} + x_{6}) + (y_{4} + y_{5} + y_{6})

(7+4) + (2+8) + (9+3)

it’s the same as

(7 + 2 + 9) + (4 + 8 + 3)

(11 + 10 + 12)

it’s the same as

18 + 15

33

it’s the same as

33

Now let’s go to the demonstration with subtraction.

Considering the set **X = { 10 _{1}, 3_{2}, 5_{3}, 7_{4}, 2_{5}, 9_{6}, 4_{7}}** and

**Y = { 5**

_{1}, 6_{2}, 1_{3}, 4_{4}, 8_{5}, 3_{6}, 10_{7}}(x_{4} – y_{4}) + (x_{5} – y_{5}) + (x_{6} – y_{6})

it’s the same as

(x_{4} + x_{5} + x_{6}) – (y_{4} + y_{5} + y_{6})

(7 – 4) + (2 – 8) + (9 – 3)

it’s the same as

(7 + 2 + 9) – (4 + 8 + 3)

(3 + (-6) + 6)

it’s the same as

18 – 15

3

it’s the same as

3

A little exercise for you to practice.

Considering the sets **X = { 5 _{1}, 7_{2}, 9_{3}, 4_{4}, 10_{5}}** and

**Y = { 2**, demonstrate that:

_{1}, 5_{2}, 1_{3}, 4_{4}, 7_{5}}### Summation of one step

Let X be a set of real values. The property states that:

The sum of a term whose start and end indexes are the same

is equal to the term itself in that index.

This one is obvious and very easy, but let’s demonstrate it so as not to lose the habit.

Considering the set **X = { 10 _{1}, 3_{2}, 5_{3}, 7_{4}, 2_{5}, 9_{6}, 4_{7}}**:

x_{3}

it’s the same as

x_{3}

5

it’s the same as

5

## Summation properties in polynomial functions

With respect to polynomial functions, the summation can be converted into ready-made formulas. This can greatly help in performing various algebraic operations.

### Summation of 1

The property states that:

The sum of term 1, in any range m to n

Let’s go to the demo:

1_{4} + 1_{5} + 1_{6} + 1_{7} + 1_{8}

it’s the same as

9 – 4

5

it’s the same as

5

Now let’s test your learning. Using the “summation of 1” formula, calculate the results of the following sums:

### Summation of an arithmetic progression

The property states that:

The summation where the term is the sum index itself, in a range from 1 to n

Let’s go to the demo:

1+2+3+4+5+6

it’s the same as

(6.7)/2

21

it’s the same as

21

Now let’s test your learning. Using the “summation of a progression” formula, calculate the result of the following sums:

### Square pyramidal number

The property states that:

The sum where the term is the squared sum index itself, in a range from 1 to n

Let’s go to the demo:

1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}

it’s the same as

(42.13)/6

1+4+9+16+25+36

it’s the same as

546/6

91

it’s the same as

91

Now let’s test your learning. Using the “square pyramidal number” formula, calculate the results of the following summation:

### David Santiago

Master in Systems and Computing. Graduated in Information Systems. Professor of Programming Language, Algorithms, Data Structures and Development of Digital Games.