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Introduction

In this chapter we extend the arithmetic and quadratic sequences studied in earlier grades, to geometric sequences. We also look at series, which is the summing of the terms in a sequence.

Arithmetic sequences

The simplest type of numerical sequence is an arithmetic sequence .

Arithmetic Sequence

An arithmetic (or linear ) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term

For example, 1 , 2 , 3 , 4 , 5 , 6 , ... is an arithmetic sequence because you add 1 to the current term to get the next term:

first term: 1
second term: 2=1+1
third term: 3=2+1
n th term: n = ( n - 1 ) + 1

Common difference :

Find the constant value that is added to get the following sequences and write out the next 5 terms.

  1. 2 , 6 , 10 , 14 , 18 , 22 , ...
  2. - 5 , - 3 , - 1 , 1 , 3 , ...
  3. 1 , 4 , 7 , 10 , 13 , 16 , ...
  4. - 1 , 10 , 21 , 32 , 43 , 54 , ...
  5. 3 , 0 , - 3 , - 6 , - 9 , - 12 , ...

General equation for the n t h -term of an arithmetic sequence

More formally, the number we start out with is called a 1 (the first term), and the difference between each successive term is denoted d , called the common difference .

The general arithmetic sequence looks like:

a 1 = a 1 a 2 = a 1 + d a 3 = a 2 + d = ( a 1 + d ) + d = a 1 + 2 d a 4 = a 3 + d = ( a 1 + 2 d ) + d = a 1 + 3 d ... a n = a 1 + d · ( n - 1 )

Thus, the equation for the n t h -term will be:

a n = a 1 + d · ( n - 1 )

Given a 1 and the common difference, d , the entire set of numbers belonging to an arithmetic sequence can be generated.

Arithmetic Sequence

An arithmetic (or linear ) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term:

a n = a n - 1 + d

where

  • a n represents the new term, the n t h -term, that is calculated;
  • a n - 1 represents the previous term, the ( n - 1 ) t h -term;
  • d represents some constant.
Test for Arithmetic Sequences

A simple test for an arithmetic sequence is to check that the difference between consecutive terms is constant:

a 2 - a 1 = a 3 - a 2 = a n - a n - 1 = d

This is quite an important equation, and is the definitive test for an arithmetic sequence. If this condition does not hold, the sequence is not an arithmetic sequence.

Plotting a graph of terms in an arithmetic sequence

Plotting a graph of the terms of sequence sometimes helps in determining the type of sequence involved.For an arithmetic sequence, plotting a n vs. n results in:

Geometric sequences

Geometric Sequences

A geometric sequence is a sequence in which every number in the sequence is equal to the previous number in the sequence, multiplied by a constant number.

This means that the ratio between consecutive numbers in the geometric sequence is a constant. We will explain what we mean by ratio after looking at the following example.

Example - a flu epidemic

What is influenza?

Influenza (commonly called “the flu”) is caused by the influenza virus, which infects the respiratory tract (nose, throat, lungs). It can cause mild to severeillness that most of us get during winter time. The main way that the influenza virus is spread is from person to person in respiratory droplets of coughs and sneezes. (This is called “dropletspread”.) This can happen when droplets from a cough or sneeze of an infected person are propelled (generally, up to a metre) through the air and deposited on the mouth or nose of people nearby. Itis good practise to cover your mouth when you cough or sneeze so as not to infect others around you when you have the flu.

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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