# Arithmetic & Geometric sequences, recursive formulae

 Page 1 / 2

## 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 $⋮$ ${n}^{\mathrm{th}}$ term: $n=\left(n-1\right)+1$

## 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 ${n}^{th}$ -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:

$\begin{array}{ccc}\hfill {a}_{1}& =& {a}_{1}\hfill \\ \hfill {a}_{2}& =& {a}_{1}+d\hfill \\ \hfill {a}_{3}& =& {a}_{2}+d=\left({a}_{1}+d\right)+d={a}_{1}+2d\hfill \\ \hfill {a}_{4}& =& {a}_{3}+d=\left({a}_{1}+2d\right)+d={a}_{1}+3d\hfill \\ \hfill ...\\ \hfill {a}_{n}& =& {a}_{1}+d·\left(n-1\right)\hfill \end{array}$

Thus, the equation for the ${n}^{th}$ -term will be:

${a}_{n}={a}_{1}+d·\left(n-1\right)$

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:

$\begin{array}{c}\hfill {a}_{n}={a}_{n-1}+d\end{array}$

where

• ${a}_{n}$ represents the new term, the ${n}^{th}$ -term, that is calculated;
• ${a}_{n-1}$ represents the previous term, the ${\left(n-1\right)}^{th}$ -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.

## 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.

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
Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers! By Jemekia Weeden By Alec Moffit By Janet Forrester By Keyaira Braxton By OpenStax By Cath Yu By Madison Christian By By Lakeima Roberts By OpenStax