# 13.1 Arithmetic and geometric sequences

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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”

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Almas
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Joseph
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Lohitha
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William
currently
William
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
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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
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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
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Damian
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Professor
I think
Professor
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Alexandre
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Alexandre
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Rafiq
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Damian
How we are making nano material?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
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Rafiq
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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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
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