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What are sequences?

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[MUSIC]

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A sequence is a list of numbers.
For example here is a sequence.

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This is a sequence that goes to one, one,
two, three, five, eight

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and I'll write dot, dot, dot to remind me
the sequence goes on forever.

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It's an unending list of numbers.
Each of these numbers in the list

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is referred to as a term in the sequence.
I want some notation so I

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can refer to specific numbers in the list.

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I'll write a sub one for the first term in
my sequence,

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a sub two for the next term, a sub three
for the next

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term, a sub four for the next term, a sub
five for

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the next term, a sub six for the next
term, and so on.

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So in this particular example, I could say
that the

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sixth term in my sequence is eight, or the
third

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term in my sequence is two.

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If I want to talk about the sequence as a
whole,

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I'll perhaps just write down a sub n,
maybe in parentheses.

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And I'll use this notation to talk about
the entire sequence.

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But by plugging in different values for n,
I

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can then speak of specific terms in the
sequence.

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This means that a sequence really amounts
to some assignment from

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these indices, the subscripts, to other
real numbers, and a gadget that

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assigns numbers to other numbers, it's a
function, so yeah the sequence

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a sub n is secretly a function, f of n,
because both

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of these gadgets are just ways of
assigning real numbers to other numbers.

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In this case, the sequence assigns a real
number to these ns, to one, two, three,

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and so on.

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So it's worth pointing out then that
what's really the

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domain of this function that's playing the
role of the sequence?

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Well, the domain of f should just be whole
numbers.

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I don't want to talk about the 5 thirds
term of the sequence, right?

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It doesn't really make sense, usually, to
talk about a sub 5 thirds, or a sub pi.

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But it does make sense to talk about a sub

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three, and a sub 100.

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So the domain of my function, the thing
that I'm allowed

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to plug in for n, should really just be
whole numbers.

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And I usually develop that with this
fancy-looking N.

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I can take a look at a sequence
numerically.

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I can just take a look at the terms and
see what their approximate values are.

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Besides thinking numerically, I could also
think about a sequence geometrically.

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For example,

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here's a number line.

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Let's say, here is zero, here is one, here
is

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two, here is three and it would keep on
going.

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And I could plot the terms of my sequence
on this number line.

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You could imagine some sequence where the
first term is here between zero and

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one, the second term is between one and
two, may be the third term.

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Is here between one and the second term.

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Maybe the

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fourth term is over here between a sub one
and one.

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Maybe the fifth term is over here, just a
little bit less than three.

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You could plot the terms in your sequence
to try

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to get some idea about what the sequence
look like.

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I can think about a sequence.

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Algebraically, well algebraically I could
define a sequence by

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giving you a rule to compute the nth term.

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May be a sub n is some algebraic

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formula like n square plus 1 and in that
case I can just compute the first term is

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1 squared plus 1 which is 2, the second
term Is 2 squared plus 1 which is 5.

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The 3rd term is 3 squared plus 1 which is
10, and so on.

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Our goal now is to build some interesting

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sequences, and explore the properties that
those sequences have.

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[NOISE]

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[NOISE]