# Introduction

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## Introduction

In earlier grades you saw patterns in the form of pictures and numbers. In this chapter, we learn more about the mathematics of patterns. Patterns are recognisable as repetitive sequences and can be found in nature, shapes, events, sets of numbers and almost everywhere you care to look. For example, seeds in a sunflower, snowflakes, geometric designs on quilts or tiles, the number sequence $0;4;8;12;16;\mathrm{...}$ .

## Investigation : patterns

Can you spot any patterns in the following lists of numbers?

1. $2;4;6;8;10;...$
2. $1;2;4;7;11;...$
3. $1;4;9;16;25;...$
4. $5;10;20;40;80;...$

## Common number patterns

Numbers can have interesting patterns. Here we list the most common patterns and how they are made.

Examples:

1. $1;4;7;10;13;16;19;22;25;...$ This sequence has a difference of 3 between each number. The pattern is continued by adding 3 to the last number each time.
2. $3;8;13;18;23;28;33;38;...$ This sequence has a difference of 5 between each number. The pattern is continued by adding 5 to the last number each time.
3. $2;4;8;16;32;64;128;256;...$ This sequence has a factor of 2 between each number. The pattern is continued by multiplying the last number by 2 each time.
4. $3;9;27;81;243;729;2187;...$ This sequence has a factor of 3 between each number. The pattern is continued by multiplying the last number by 3 each time.

## Triangular numbers

$1;3;6;10;15;21;28;36;45;...$

This sequence is generated from a pattern of dots which form a triangle. By adding another row of dots (with one more dot in each row than in the previous row) and counting all the dots, we can find the next number of the sequence.

## Square numbers

$1;4;9;16;25;36;49;64;81;...$

The next number is made by squaring the number of the position in the pattern. The second number is 2 squared ( ${2}^{2}\phantom{\rule{3.33333pt}{0ex}}or\phantom{\rule{3.33333pt}{0ex}}2×2$ ). The seventh number is 7 squared ( ${7}^{2}\phantom{\rule{3.33333pt}{0ex}}or\phantom{\rule{3.33333pt}{0ex}}7×7$ ) etc.

## Cube numbers

$1;8;27;64;125;216;343;512;729;...$

The next number is made by cubing the number of the position in the pattern. The second number is 2 cubed ( ${2}^{3}\phantom{\rule{3.33333pt}{0ex}}or\phantom{\rule{3.33333pt}{0ex}}2×2×2$ ). The seventh number is 7 cubed ( ${7}^{3}\phantom{\rule{3.33333pt}{0ex}}or\phantom{\rule{3.33333pt}{0ex}}7×7×7$ ) etc.

## Fibonacci numbers

$0;1;1;2;3;5;8;13;21;34;...$

The next number is found by adding the two numbers before it together. The 2 is found by adding the two numbers in front of it ( $1+1$ ). The 21 is found by adding the two numbers in front of it ( $8+13$ ). The next number in the sequence above would be 55 ( $21+34$ ).

Can you figure out the next few numbers?

Say you and 3 friends decide to study for Maths, and you are seated at a square table. A few minutes later, 2 other friends join you and would like to sit at your table and help you study. Naturally, you move another table and add it to the existing one. Now 6 of you sit at the table. Another 2 of your friends join your table, and you take a third table and add it to the existing tables. Now 8 of you can sit comfortably.

Examine how the number of people sitting is related to the number of tables.

1.  Number of Tables , $n$ Number of people seated 1 $4=4$ 2 $4+2=6$ 3 $4+2+2=8$ 4 $4+2+2+2=10$ $⋮$ $⋮$ $n$ $4+2+2+2+...+2$
2. We can see that for 3 tables we can seat 8 people, for 4 tables we can seat 10 people and so on. We started out with 4 people and added two each time. Thus, for each table added, the number of persons increased by 2.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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