# 13.1 Arithmetic and geometric sequences

 Page 1 / 1
A teacher's guide for lecturing on arithmetic and geometric sequences.

The in-class assignment does not need any introduction. Most of them will get the numbers, but they may need help with the last row, with the letters.

After this assignment, however, there is a fair bit of talking to do. They have all the concepts; now we have to dump a lot of words on them.

A “sequence” is a list of numbers. In principal, it could be anything: the phone number 8,6,7,5,3,0,9 is a sequence.

Of course, we will not be focusing on random sequences like that one. Our sequences will usually be expressed by a formula: for instance, “the xxxnth terms of this sequence is given by the formula $100+3\left(n-1\right)$ ” (or $3n+97\right)$ in the case of the first problem on the worksheet. This is a lot like expressing the function $y=100+3\left(x-1\right)$ , but it is not exactly the same. In the function $y=3x+97$ , the variable $x$ can be literally any number. But in a sequence , xxxn must be a positive integer; you do not have a “minus third term” or a “two-and-a-halfth term.”

The first term in the sequence is referred to as ${t}_{1}$ and so on. So in our first example, ${t}_{5}=112$ .

The number of terms in a sequence, or the particular term you want, is often designated by the letter $n$ .

Our first sequence adds the same amount every time. This is called an arithmetic sequence . The amount it goes up by is called the common difference $d$ (since it is the difference between any two adjacent terms). Note the relationship to linear functions, and slope.

If I want to know all about a given arithmetic sequence, what do I need to know? Answer: I need to know ${t}_{1}$ and $d$ .

OK, so if I have ${t}_{1}$ and $d$ for the arithmetic sequence, give me a formula for the ${n}^{\mathrm{th}}$ term in the sequence. (Answer: ${t}_{n}={t}_{1}+d\left(n-1\right)$ . Talk through this carefully before proceeding.)

Time for some more words. A recursive definition of a sequence defines each term in terms of the previous. For an arithmetic sequence, the recursive definition is ${t}_{n+1}={t}_{n}+d$ . (For instance, in our example, ${t}_{n+1}={t}_{n}+3.\right)$ An explicit definition defines each term as an absolute formula, like the $3n+97$ or the more general ${t}_{n}={t}_{1}+d\left(n-1\right)$ we came up with.

Our second sequence multiplies by the same amount every time. This is called a geometric sequence . The amount it multiplies by is called the common ratio $r$ (since it is the ratio of any two adjacent terms).

Find the recursive definition of a geometric sequence. (Answer: ${t}_{n+1}=r{t}_{n}$ . They will do the explicit definition in the homework.)

Question: How do you make an arithmetic sequence go down ? Answer: $d<0$

Question: How do you make a geometric series go down? Answer: $0 . (Negative $r$ values get weird and interesting in their own way...why?)

## Homework

“Homework: Arithmetic and Geometric Sequences”

#### Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
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
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Advanced algebra ii: teacher's guide. OpenStax CNX. Aug 13, 2009 Download for free at http://cnx.org/content/col10687/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Advanced algebra ii: teacher's guide' conversation and receive update notifications? By Angela January By Jonathan Long By Sebastian Sieczko... By Madison Christian By P. Wynn Norman By Samuel Madden By Danielrosenberger By Anonymous User By OpenStax By Alec Moffit