What I want to do in this video is familiarize ourselves
with a very common class of sequences.
And this is arithmetic sequences.
And they are usually pretty easy to spot.
They are sequences where each term is a fixed number larger
than the term before it.
So my goal here is to figure out which of these sequences
are arithmetic sequences.
And then just so that we have some practice
with some of the sequence notation,
I want to define them either as explicit functions
of the term you're looking for, the index you're looking at,
or as recursive definitions.
So first, given that an arithmetic sequence
is one where each successive term is a fixed amount larger
than the previous one, which of these are arithmetic sequences?
Well let's look at this first one right over here.
To go from negative 5 to negative 3, we had to add 2.
Then to go from negative 3 to negative 1, you have to add 2.
Then to go from negative 1 to 1, you had to add 2.
So this is clearly an arithmetic sequence.
We're adding the same amount every time.
And there are several ways that we could define the sequence.
We could say it's a sub n.
And you don't always have to use k.
This time I'll use n to denote our index.
From n equals 1 to infinity with-- and there's two ways
we could define it.
We could either define it explicitly,
or we could define it recursively.
So if we wanted to define it explicitly,
we could write a sub n is equal to whatever the first term is.
In this case, our first term is negative 5.
It's equal to negative 5 plus-- we're
going to add 2 one less times than the term we're at.
So for the second term, we add 2 once.
For the third term, we add 2 twice.
For the fourth term, from our base term,
we added 2 three times.
So we're going to add 2.
We're going to add positive 2 one less than the index
that we're looking at-- n minus 1 times.
So this is an explicit definition
of this arithmetic sequence.
If I wanted to write it recursively,
I could say a sub 1 is equal to negative 5.
And then each successive term, for a sub 2 and greater--
so I could say a sub n is equal to a sub n minus 1 plus 3.
Each term is equal to the previous term-- oh, not 3--
plus 2.
So this is for n is greater than or equal to 2.
So either of these are completely
legitimate ways of defining the arithmetic sequence
that we have here.
We can either define it explicitly,
or we could define it recursively.
Now let's look at this sequence.
Is this one arithmetic?
Well, we're going from 100.
We add 7.
107 to 114, we're adding 7.
114 to 121, we are adding 7.
So this is indeed an arithmetic sequence.
So just to be clear, this is one,
and this is one right over here.
And we could write that this is the sequence a sub n,
n going from 1 to infinity of-- and we could just
say a sub n, if we want to define it explicitly,
is equal to 100 plus we're adding 7 every time.
And then each term-- the second term we added 7 once.
Third term-- we add 7 twice.
So for the nth term, we're going to add 7 n minus 1 times.
So this is an explicit definition of it,
but we could also do it recursively.
So just to be clear, this is one definition where we write it
like this, or we could write a sub n,
from n equals 1 to infinity.
And in either case I should write with.
And if I want to define it recursively,
I could say a sub 1 is equal to 100.
And then, for anything larger than 1, for any index above 1,
a sub n is equal to the previous term plus 7.
And so we're done.
This is another way of defining it.
So in general, if you wanted a generalizable way
to spot or define an arithmetic sequence,
you could say an arithmetic sequence
is going to be of the form a sub n-- if we're talking
about an infinite one-- from n equals 1 to infinity.
If you want to define it explicitly,
you could say a sub n is equal to some constant, which
would essentially the first term.
It would be some constant plus some number
that your incrementing-- or I guess
this could be a negative number, or decrementing
by-- times n minus 1.
So this is one way to define an arithmetic sequence.
In this case, d was 2.
In this case, d is 7.
That's how much you're adding by each time.
And in this case, k is negative 5, and in this case, k is 100.
The other way, if you wanted to the right
the recursive way of defining an arithmetic sequence generally,
you could say a sub 1 is equal to k,
and then a sub n is equal to a sub n minus 1.
A given term is equal to the previous term
plus d for n greater than or equal to 2.
So once again, this is explicit.
This is the recursive way of defining it.
And we would just write with there.
Now the last question I have is is this one right over here
an arithmetic sequence?
Well, let's check it out.
We start at 1.
Then we add 2.
Then we add 3.
So this is an immediate giveaway that this is not
an arithmetic sequence.
Now we are adding 4.
We're adding a different amount every time.
So this, first of all, this is not arithmetic.
This is not an arithmetic sequence.
But how could we define this, since we're
trying to define our sequences?
Let's say we wanted to define it recursively.
So we could say, this is equal to a sub n, where
n is starting at 1 and it's going to infinity,
with-- we'll say our base case-- a sub 1 is equal to 1.
And then for n is 2 or greater, a sub n
is going to be equal to what?
So a sub 2 is the previous term plus 2. a sub 3
is the previous term plus 3. a sub 4
is the previous term plus 4.
So it's going to be the previous term plus whatever
your index is.
So this looks close, but notice here we're
changing the amount that we're adding
based on what our index is.
We're adding the amount of index to the previous term.
And so this is for n is greater than or equal to 2.
Well for an arithmetic sequence, we're
adding the same amount regardless
of what our index is.
Here we're adding the index itself.
So this one is not arithmetic, but it's
an interesting sequence nonetheless.