What are sequences?
[MUSIC]
A sequence is a list of numbers. For example here is a sequence.
This is a sequence that goes to one, one, two, three, five, eight
and I'll write dot, dot, dot to remind me the sequence goes on forever.
It's an unending list of numbers. Each of these numbers in the list
is referred to as a term in the sequence. I want some notation so I
can refer to specific numbers in the list.
I'll write a sub one for the first term in my sequence,
a sub two for the next term, a sub three for the next
term, a sub four for the next term, a sub five for
the next term, a sub six for the next term, and so on.
So in this particular example, I could say that the
sixth term in my sequence is eight, or the third
term in my sequence is two.
If I want to talk about the sequence as a whole,
I'll perhaps just write down a sub n, maybe in parentheses.
And I'll use this notation to talk about the entire sequence.
But by plugging in different values for n, I
can then speak of specific terms in the sequence.
This means that a sequence really amounts to some assignment from
these indices, the subscripts, to other real numbers, and a gadget that
assigns numbers to other numbers, it's a function, so yeah the sequence
a sub n is secretly a function, f of n, because both
of these gadgets are just ways of assigning real numbers to other numbers.
In this case, the sequence assigns a real number to these ns, to one, two, three,
and so on.
So it's worth pointing out then that what's really the
domain of this function that's playing the role of the sequence?
Well, the domain of f should just be whole numbers.
I don't want to talk about the 5 thirds term of the sequence, right?
It doesn't really make sense, usually, to talk about a sub 5 thirds, or a sub pi.
But it does make sense to talk about a sub
three, and a sub 100.
So the domain of my function, the thing that I'm allowed
to plug in for n, should really just be whole numbers.
And I usually develop that with this fancy-looking N.
I can take a look at a sequence numerically.
I can just take a look at the terms and see what their approximate values are.
Besides thinking numerically, I could also think about a sequence geometrically.
For example,
here's a number line.
Let's say, here is zero, here is one, here is
two, here is three and it would keep on going.
And I could plot the terms of my sequence on this number line.
You could imagine some sequence where the first term is here between zero and
one, the second term is between one and two, may be the third term.
Is here between one and the second term.
Maybe the
fourth term is over here between a sub one and one.
Maybe the fifth term is over here, just a little bit less than three.
You could plot the terms in your sequence to try
to get some idea about what the sequence look like.
I can think about a sequence.
Algebraically, well algebraically I could define a sequence by
giving you a rule to compute the nth term.
May be a sub n is some algebraic
formula like n square plus 1 and in that case I can just compute the first term is
1 squared plus 1 which is 2, the second term Is 2 squared plus 1 which is 5.
The 3rd term is 3 squared plus 1 which is 10, and so on.
Our goal now is to build some interesting
sequences, and explore the properties that those sequences have.
[NOISE]