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So Marco has a collection of 437 antique soda bottles and each year he buys 32
more. And we want to know how many bottles
he'll have in five years or how long it will take for his collection to reach
1,000 bottles. And to do this we're going to need some
notation. And so we're going to start out by
introducing a subscript base notation for a population or the amount of bottles
that he has. We're going to use the term population a
lot because we're going to be talking about populations.
In this case it's a population of bottles.
So we're going to use P to represent the population or the number of bottles and
we're going to subscript it with an n. So this would be the number of bottles
that Marco has after n years. So the subscript lets us keep track of
which year we're talking about. So P, 0 would be how many bottles he has
after 0 years in other words right now. How many does he have?
He has 437 bottles right now and so P, 0 is 437.
Now if we wanted to know how many he'll have in year 1, or in other words after 1
year We could notice that each year he buys 32 more so that's 437 plus 32.
And 437 plus 437 plus 32 is 469 now. Notice where this value came from.
This was my previous year's amount, so we could in fact replace this with p sub 0
and we could say p1 is p0 plus 32. Now notice, my next year in year two I'd
have 469 bottles from the previous year and I've added 32 more to end up with
501, but where did this 469 come from. That was my previous year.
That was my p sub one. So each year I can figure out how many
bottles I have by adding 32 to the previous year's number of bottles.
And I can generalize this. I can generalize this ideal to say,
listen, I start with 437 bottles, and in any given year the number of bottles I
have will be the number I had in the previous year, so n minus 1 is the year
before. Plus an additional 32.
So for example, p3 would be p2, right? because 3 minus 1 is 2, plus 32.
I already know what p2 is, p2 is 501. Add 32.
So P 3 is 533 bottles. And this equation that we've come up
with, is called a recursive equation. This is a recursive, re Recursive
equation. A recursive equation is one where the
next value depends upon the previous value.
And in general, this type of growth that we're looking at here is called linear
growth. So this is linear growth.
Linear growth is what we get whenever we grow by a constant.
When we grow by a constant number amount. So in this case we're growing by 32 each,
each year. So when we have a recursive linear
growth. We can always write the equation in the
form p, n, so the population year n is always going to be the population in the
previous year plus some d. Where this is the, what we call a common
difference, in other words, it's the amount that we increase by, each year.
In this case, this value, 32, is my d, it is my common difference.
And so this is the general form of a recursive equation.
We always have to have our initial amount, our P sub 0, and then we have a
recursive equation, that will look like this.