Expression for compound or exponential growth

You put $3,800 in a savings account. The bank will provide 1.8% interest on the money in the account every year. Another way of saying that is that the money in the savings account will grow by 1.8% per year.

Write an expression that describes how much money will be in the account in 15 years. So, let's just think about this a little bit. Let's just think about the starting amount.

So, in the start, we're just going to put $3,800. We could view that as year zero. Year—actually let me write it that way. So the start is the same thing as year zero, and we're going to start with $3,800.

Now, let's think about year one. How much money will we have after one year? Well, we would have the original amount that we put, $3,800, and then we're going to get the amount that we get in interest. They say that the bank will provide 1.8% interest on the money in the account, so it'll be plus 1.8% times $3,800.

We could also write this as a decimal. This is equal to $3,800 plus—I'll just write—I'll switch the order of multiplication here. Plus $3,800 times 0.018.

1.8% is the same thing as 18,000 or 1.8 hundredths, depending on how you want to pronounce it. And so here, you might say, "Well, there's kind of an interesting potential simplification mathematically here." I could factor $3,800 out of each of these terms. I have a $3,800 here; I have a $3,800 here.

So why don't I factor it out—essentially undistributed it? This is going to be $3,800 times—when you factor it out here, you get a 1—plus—when you factor it out here, you get 0.018. So I could just rewrite this as $3,800 times 1.018.

So this is an interesting time to pause. We're not at the full answer yet—how much we have in 15 years—but we have an interesting expression for how much we have after one year. Notice that if the money is growing by 1.8%, or another way it was growing by 0.018, that's equivalent to multiplying the amount that we started the year with by one plus the amount that it's growing by, or 1.018.

And once again, why does this make intuitive sense? Because at the end of the year, you're going to have the original amount that you put—that's what that one really represents—and then plus you're going to have the amount that you grew by. So you multiply both the sum here times the original amount you put, and that's how much you'll have at the end of year 1.

What about year two? So year two—well, we know what we're going to start with in year two. We're going to start with whatever we finished year one with, so we're going to start with $3,800 times 1.018. But then it's going to grow by 1.8%, or grow by 0.018.

And we already said if you're going to grow by that amount, that's equivalent to multiplying it by 1.018. Well, this is the same thing as $3,800 * 1.018 to the 2 power.

I think you see where this is going. Every time we grow by 1.8%, we're going to multiply by 1.018. If we're thinking about 15 years in the future, we're going to do that 15 times. So one year in the future, your exponent here is essentially one; two years, your exponent is two.

So year 15—I can just cut to the chase here—so year 15, well that's just going to be, we're going to have the original amount that we invested and we are going to grow 1.018 15 times. So we're going to multiply by this amount 15 times to get the final amount.

And one of the fun things—this is actually called compound growth—where every year you grow on top of the amount that you had before. You'll see if you type this into a calculator that, even though 1.8% per year does not seem like a lot, over 15 years it actually would amount to a reasonable amount.

But this is the expression. They're not asking us to calculate it; they just want us to know an expression that describes how much money will be in the account in 15 years.