Welcome to an introduction to sequences.
The goals of this video are to define a sequence,
and also to determine if a given sequence is
arithmetic or geometric.
A sequence is an ordered list of terms or elements,
as we see here.
And each term in a sequence is identified by its location
in the sequence.
So a sub 1 is the first term in the sequence,
a sub 2 would be the second term,
and a sub 3 would be the third term,
a sub 4 is the fourth term, and so on.
So the subscript tells us which term in the sequence
we're referring to.
There's also two types of sequences.
There's a finite sequence and an infinite sequence.
A finite sequence is a sequence that
has a specific number of terms as we see here,
and then it stops.
An infinite sequence, as we see here, continues on forever.
And we know that because of the dot, dot, dot on the right.
Both types of sequences can be considered a function, where
the domain would be 1, 2, 3, and so on,
referring to the position of the term in the sequence.
And the range would consist of the set of terms
that make up the sequence.
And often a sequence is expressed as a rule
or a formula, as we see here.
So the value of n is based upon the term
that we're looking for in the sequence.
So if we want a sub 1 or the first term,
that means n is equal to 1.
If we want the third term, that means we're looking for a sub
3, so n is equal to 3.
So let's go ahead and practice this.
Let's say we want to find the first four terms of these two
sequences.
That means we want to find a sub 1, a sub 2, a sub 3,
and a sub 4.
Well, if you want to find a sub 1, that means n is equal to 1.
So we have 3 times 1, minus 5.
So the first term is equal to negative 2.
a sub 2 is telling us that now n is equal to 2.
So we have 3 times 2, minus 5.
That'll 6 minus 5, which is equal to 1.
And we want to find a sub 3.
So now n is equal to 3.
So we have 3 times 3, minus 5.
9 minus 5 is equal to 4.
And then lastly, a sub 4 means n is equal to 4.
So 3 times 4, minus 5.
That'll be 12 minus 5, which is equal to 7.
So the first four terms of the sequence
are negative 2, 1, 4, and 7.
So in this case, the domain would be 1, 2, 3, 4.
And the range would be negative 2, 1, 4, and 7.
Let's go ahead and try this again.
We want to find a sub 1, a sub 2, a sub 3, and a sub 4.
So notice n is now the exponent.
So we have to the first, plus 1, for a sub 1.
That would be 2 plus 1, or 3.
For a sub 2, we'll have 2 do the second power, plus 1.
That's 4 plus 1 or 5.
And for a sub 3, we have 2 to the power of 3, plus 1.
That would be 8 plus or 9.
And then lastly, a sub 4 would be 2 the fourth, plus 1.
2 to the fourth is 16, plus 1.
So we have 17.
So the first four terms of this sequence
would be 3, 5, 9, and 17.
Another way to express a sequence
is by a recursive formula.
And this means that each term in a sequence
is based upon previous terms, not the value of n.
So if you look at this formula here that generates the terms,
if we want to find let's say a sub 3,
we have to find a sub 3 minus 1 or a sub 2.
So a sub 3 is based upon the value of a sub 2.
If we wanted a sub 5, then this would be a sub 5 minus 1,
or a sub 4.
The fifth term is based upon the value of the fourth term.
So notice that they give us some extra information here
to get us started.
We know that a sub 1 is equal to 3.
Let's go ahead and find the next two terms.
So the first term is equal to 3.
Let's see if we can find the next two terms.
So if we want to find a sub 2, we know n is equal to 2.
So we're going to have 4 times a sub--
well, 2 minus 1 would be 1.
So we have a sub 1, minus 3.
We know the value of a sub 1.
It was given to us as 3.
So we have 4 times 3, minus 3.
So we have 12 minus 3, which is equal to 9.
The second term is equal to 9.
Let's go and find one more term.
a sub 3 is going to be equal to--
well, now n is equal to 3.
So now we have 4 times a sub 3 minus 1.
That'll be a sub 2, minus 3.
That'll give us 4 times a sub 2, which we just
found is equal to 9, and then minus the constant, 3.
So here we have 4 times 9 is 36, minus 3.
So the next term is equal to 33 So
as you can see with this process,
if we wanted to find, let's say, the fifth term, we would have
to keep going here and find the fourth term in order
to find the fifth term.
The second example of a recursive formula
actually requires two previous terms.
And you can see, they give us the first two terms to start.
Let's go ahead and give it a try.
So we know that the first two terms are 1 and 2.
So the next term would be a sub 3.
So now n is 3.
So we have a sub 3 minus 2, that's a sub 1, plus a sub--
again, n is still 3.
So 3 minus 1 would be a sub 2.
Well, a sub 1 is equal to 1, plus a sub
2, which is equal to 2.
So we have 3.
a sub 3 is equal to 3.
Let's go ahead and find one more term.
a sub 4 is going to equal a sub 4 minus 2,
that's a sub 2, plus a sub 4 minus 1, that's a sub 3.
So a sub 4 is equal to a sub 2, which is 2, plus a sub
3, which we just found is 3.
So it's equal to 5.
So that's how a recursive formula works.
Now we're going to shift gears to talk
about two different types of sequences.
One's an arithmetic sequence and the other
is a geometric sequence.
An arithmetic sequence is a sequence
that has the pattern of adding the same value to determine
consecutive terms.
And we say arithmetic sequences have a common difference.
So if you take a look at the sequence,
and start looking for a pattern, what you'll notice
is that if we subtract 6 each time,
we can generate these terms in the sequence.
Well, the definition says add the same value each time.
So remember instead of subtracting 6,
we could add a negative 6.
And so the common difference in the sequence is negative 6.
Another way to determine the common difference
of an arithmetic sequence is to take any term in the sequence
and subtract the term before it.
The way we express that is, the difference is equal to a sub
n minus a sub n minus 1.
So this is the fourth term.
This would be 4 minus 1, the third term.
So we can pick any term and subtract the one before it.
Of course, we can't start with the first term though.
So if we start with a sub 2, we would just subtract a sub 1
to find the common difference.
So we'd have 31 minus 37, which equals negative 6.
And again, we can pick any term and subtract the term before.
For a geometric sequence, the pattern
is that we multiply by a constant
to determine consecutive terms.
And we say geometric sequences have a common ratio, rather
than a common difference.
And we can calculate that by taking
any term in the sequence, and dividing by the term before it.
If we take a look at this sequence here,
you may notice a pattern.
And again, there's a couple of ways to identify it.
To go from 9 to 3 and then 3 to 1,
you could say, well, the pattern is dividing by 3.
And that is true.
But remember that dividing by 3 is
the same as multiplying by 1/3.
So in this case, our r or common ratio is equal to 1/3.
And if we want to calculate it using
the terms in the sequence, again, we
can pick any term and divide by the term before it.
So if we want to use the second term,
we would just divide by the first term.
And so we'd have 3 divided by 9, which is equal to 1/3.
This ratio will be the same regardless of which two terms
we use.
OK, we'll take a closer look at both the arithmetic
and geometric sequences in the next couple of videos.
Thank you for watching.