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So based on this percentage growth scenario we came up with this recursive
model for our growth. And let's see if now we can come up with
an explicit equation because this recursive equation, just like every
recursive equation, is kind of hard to use to actually make calculations.
And so let's go back to our specific scenario.
So we said that P 1 was going to be 1.10 times P0, based on our recursive
equation. So, likewise, P2 would be 1.10 times P1.
But we know what p1 is, right? So let's go ahead and replace p1 with
what we know is equivalent to p1 is 1.10 times p0.
Now can we simplify this. at all.
Why yes I can say 1.10 times 1.10 is 1.10 squared, times p0.
So now same idea. P3 is 1.10 times p2, but I know something
about p2. I know that P2 is 1.10 squared time P0,
right? That was P2 is 1.10 squared times P0.
And now 1.10 times 1.010 squared is 1.10 cube times P0 and you probably see the
trend that's going to continue here. If I continue this down in general pn is
going to be 1.10 raised to the power of n times p zero.
And this is going to give me my, explicit equation.
So lets see if we can generalize off of this.
So P n is going to be, lets see, where did this 1.10 come from?
That was the 1 plus r, right? That's 1 plus r, where again r is the
Growth rate and that's a percentage growth rate.
Raised to the N power. Times P0 and this is the explicit
equation for what is called exponetial growth and you can see now why it's
called exponential growth, it's because in this type of equation, the variable is
in the exponent and hence exponential spell it right this time, exponential
growth. Now, notice this is different than
something like a quadratic, like x squared, where the variable is in the
base being raised to a numerical power. In this case, the variable is in fact in
the exponent. Now we can go ahead and use this equation
to make predictions. So for example, if I wanted to predict
the population, the population in, let's say, 20 years, I could figure out.
what P sub 20 is, that would be 1.10 to the 20 times the initial population of
1,000. Now this is going to require me to
calculate 1.10 to the 20th power, which I mean admittedly I could do by multiplying
1.10 times itself 20 times, but that'd be really tedious, and so this is a good
time to pull out our calculators. So many calculators, scientific
calculators, will have some sort of button on them like x to the y or y to
the x or some just have a carrot key. That's that vertical up bar thing.
and so to Evaluate this power we'll say 1.10 raised to the power of 20.
And now I'm going to go ahead and hit equal to figure out what that is so that
is 1.10 to the 20th and now I'm going to multiply that by my P0 by my 1000.
Gives me 6727 as my prediction. So that's 6727 approximately, as the
prediction for the population in 20 years.
Using this exponential growth model.