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ALBERT MEYER: We've considered
arithmetic sums, where

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each term is a fixed amount
larger than the previous term

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by an additive amount.

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Another kind of sum
called geometric sums

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which, we're going
to look at now,

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come about where
each sum is a fixed

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multiple of the previous sum.

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And these come up all the time
in lots of different settings.

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And in this
particular case, we're

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going to look at
an example of how

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it applies to analyzing the
value of money in the future.

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So let's begin with
a geometric sum.

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And there's the
standard form of it.

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Geometric sum is of the form
1 plus x plus x squared up

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through the nth power of x.

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And, for uniformity, notice
that 1 is actually x to the 0.

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So we're taking the sum from
k equals 0 to n of x to the k.

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Now, what I'd like to do is
find a nice closed form for this

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without those
ellipses, and having

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a growing number of terms n.

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And there's a simple trick
with the arithmetic sum.

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The way Gauss got
it and the way we

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got it was by reversing the sum
and adding the things together.

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This time the trick is
to multiply Gn by x.

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Now what's that
going to do is it's

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going to increase the
power of x in each term

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by one, which is tantamount
to a right shift.

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Let's look at that.

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There's xG of n, so
1 times x, course,

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I'm subtracting looking ahead.

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So 1 times x is minus
x times x is x squared.

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I'm going to subtract
it down through x

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to the n minus 1 times x gives
me the x to the nth term.

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And finally, I
have an extra term

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from right shifting x to the
n to be x to the n plus 1.

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Now let's do this subtraction.

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But of course, I'm going
to align the terms up so

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that they're easy to subtract.

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And now all the terms
in the middle cancel,

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which is very cool,
because we've just

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figured out that Gn minus xGn
is 1 minus x to the n plus 1.

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So we have this nice,
elegant formula.

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Now I can factor out Gn
and I get a Gn times 1

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minus x on the left.

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And the result is that I get
the Gn is 1 minus x to the n

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plus 1 over 1 minus x.

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This is actually a formula that
we proved before by induction.

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But when we did it
by induction, there

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was no clue about who
was the clever person

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to think of this formula.

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Now you know how that
clever person found it.

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And this is kind
of a standard trick

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that we'll see more of when we
look at generating functions.

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But for now, it's a
simple trick for getting

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a nice closed form for a sum.

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We refer to it as the
perturbation method.

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You take the sum, you
perturb it a little,

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see how it relates to itself,
get an arithmetic relation,

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and solve for a
formula for the sum.

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OK.

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A geometric series--
I use the word

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sum for a finite sum--
a geometric series

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is when you take
an infinite sum.

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So the infinite geometric
sum is the sum 1

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plus x plus x
squared x to the n.

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And it keeps going.

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It's the sum from i equals
0 to infinity of x to the i.

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And there's a simple
formula for that too.

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It's even simpler actually.

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Because the definition of
an infinite sum is it's

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the limit of the truncated sums.

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It's the limit of the
sum of the first n terms

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as n goes to infinity,
assuming that limit exists.

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And so for the value
of this infinite series

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is the limit of Gn,
which is the sum up to n,

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and let's look at that.

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Well, Gn is 1 minus x to
the n plus 1 over 1 minus x.

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So the limit of that, the limits
distribute pass the 1 down to--

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and the x doesn't
have an n in it,

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so it winds up being 1 minus the
limit as n approaches infinity

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of x to the n plus 1
divided by 1 minus x.

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And as long as x is
less than 1, x to the n

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plus 1 is going to go
to 0, and I wind up

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with this nice simple formula
that the infinite series,

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the sum from i equals 0
to infinity of x to the i

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is equal to 1 over
1 minus x, providing

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that the magnitude
of x is less than 1.

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OK.

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That's the basics mathematical
preliminaries of geometric sums

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and geometric series.

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Now, let's look at a
typical application

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having to do with the
future value of money.

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Suppose we want to make
the following deal,

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I promise I will pay
you $100 in one year

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if you will pay me
a fixed amount now.

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So let's call it x dollars.

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And the puzzle is how
much money is $100 worth

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if you can't have it now?

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You can only have
it in one year.

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It's worth x dollars.

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How are we going to figure
out what x should be?

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What would be a fair
amount for you to pay me

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so that I'm willing to pay you
$100 in one year and it's fair,

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nobody loses?

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OK.

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Well, here's the basic fact
that is the basis for evaluating

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what the value of money
in the future is is I'm

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going to assume that my bank
will pay me 3% interest.

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This is a generous bank
in today's economy,

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but it used to be
a stingy offer.

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Interest rates in my lifetime
have ranged between about 17%

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a year down to less
than 1% a year.

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3% is a reasonable
number to play with.

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So let's suppose that my
bank commits to paying me

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3% interest on a deposit now.

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That is to say, let's define
the bank rate, b, to be 1.03.

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And the deal is
that the bank will

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increase the money
that I have now

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by a factor of b in one year.

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OK.

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Well, so if I deposit
your x dollars now,

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that means I will have b
times x dollars in one year.

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OK.

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Assuming that the bank
is completely reliable,

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there's no risk there and I get
exactly b times x in one year,

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then I won't lose
any money providing

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that b times x is greater
than or equal to 100.

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I need the x dollars
you give me now

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to be worth the $100
I'm supposed to pay you.

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I'll come out ahead if bx is
greater than or equal to 100.

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I'll loose if it's
less than 100.

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And it's completely fair
if bx is equal to 100.

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All right.

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Well, that means
that x is simply

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100 over b, which we
decided was 1.03 $97.09.

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So $100 in one year is worth
$97.09 or normalizing to $1.00,

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$1.00 in one year is worth
$0.97 essentially now.

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Well, now we can shift
perspective a little bit

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and think back a year.

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So how much money did I need
last year in order to be worth

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$1.00 today?

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Well, by the same
reasoning, the bank

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is going to pay me
b times r today.

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So I need b times r
to equal a dollar.

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In other words, r has
got to be 1 over ,

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r is 1 over the bank rate.

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So $1.00 a year ago is worth--
r dollars a year ago is worth

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a dollar today.

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And by the same reasoning,
n dollars paid in two years

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is worth n times r paid in
one year, which is worth n

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times r squared paid today.

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So I can iterate this one
over bank rate factor,

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and as time goes on, k years
out-- a value of n dollars paid

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in k years is worth n
times r to the k today,

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where r is 1 over the bank rate.

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OK.

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That's good to know.

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Let's think about annuities now.

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An annuity is a
contract that people by

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to provide income for
themselves without risk.

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So they will make
a deal typically

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with an insurance
company where they

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will pay a certain
amount of money

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now to the insurance company,
and the insurance company

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promises to provide
them a regular income

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sometimes for life or
sometimes for a fixed period.

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So let's look at an example.

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I will pay you $100
a year for 10 years

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if you will pay me
a fixed premium.

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What should it be?

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So I'm going to promise
as the insurance company

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to pay you $100 a
year for 10 years.

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I want you to pay me a
premium of y dollars now.

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How much should you pay?

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Well, let's think about it.

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$100 in one year is
worth 100 times r.

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And $100 in two years is
worth 100 times r squared.

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And finally, $100 in 10 years is
worth 100 times r to the 10th.

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So this is the
amount that I will

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have to pay you in today's
dollars-- I need-- in order

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to be paying you $100
a year for 10 years.

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I need a total of this
much amount of money

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now, because each of these
terms is the amount of money

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that the $100 will be worth
paid to you that many years out.

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Well, look at this sum.

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If I factor out 100
r, I'm left with 100

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r times the geometric sum
from 1 to r to the ninth,

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where the base of the sum
is r, the factor is r.

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Well, we have a nice
formula for that.

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It's simply 1 minus r to
the 10th over 1 minus r.

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And now plugging in
r equals 1 over 1.03.

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I wind up with the conclusion
that this annuity is

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worth $853.02 today.

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My promise to pay you $1,000,
but spread out over the next 11

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years, is worth $853.02
today, assuming that the bank

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rate is 3% a year.

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And that's a typical case
where geometric series come up.

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And you'll see other
examples in problems.

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Just a quick thing
to think about.

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Suppose that the bank rates
rapidly increase, unexpectedly

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increase.

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The Fed finally gets
the economy moving

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and interest rates
run up to 5%, say.

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What happens on this annuity?

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You've already paid me
the 853 and I've already

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committed to paying you $100
a year for the next 10 years

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starting in a year.

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Who comes out ahead if
bank rates increase?

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You come out ahead,
the deal stays fair,

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or I come out ahead.

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And I'll close by letting you
think about that question.