Geometric series introduction | Algebra 2 | Khan Academy

In this video, we're going to study geometric series. To understand that, I'm going to construct a little bit of a table to understand how our money could grow if we keep depositing, let's say, a thousand dollars a year in a bank account.

So, let's say this is the year, and we're going to think about how much we have at the beginning of the year. Then, this is the dollars in our account, and let's say that the bank is always willing to give us five percent per year, which is pretty good. It's very hard to find a bank account that will actually give you five percent growth per year.

So, that means if you put a hundred dollars in, at the end of a year, or exactly a year later, it'd be 105 dollars. If you put a thousand dollars in, a year later, it would be a thousand fifty. It'd be five percent larger. So, let's say that we want to put a thousand dollars in per year. I want to think about, well, what is going to be my balance at the beginning of year one, at the beginning of year two, at the beginning of year three, and then see if we can come up with a general expression for the beginning of year n.

So, year one, right at the beginning of the year, I put in 1,000 dollars in the account. That's pretty straightforward. But then what happens in year two? I'm going to deposit a thousand dollars, but then that original thousand dollars that I have would have grown. So, I'm going to deposit one thousand dollars, and then the original thousand dollars that I put at the beginning of year one, that is now grown by five percent. Growing by five percent is the same thing as multiplying by 1.05.

So, this is now going to be plus one thousand dollars times 1.05, fairly straightforward. Now, what about the beginning of year three? How much would I have in the bank account right when I've made that first year three deposit? Pause this video and see if you can figure that out.

Well, just like at the beginning of year two and the beginning of year one, we're going to make a thousand dollar deposit. But now the money from year two has grown by five percent. So, this is now going to be one thousand dollars times 1.05. Then, that money that we originally deposited from year one, that was one thousand times 1.05 in year two, that's going to grow by another five percent.

So, this is going to be plus 1,000 times 1.05 times 1.05. We're going by another five percent! Well, we could just rewrite this part right over here as 1.05 squared. So, do you see a general pattern that's going to happen here?

Well, as you go to year n, in fact, pause the video again and see if you could write a general expression. You're going to have to do a little bit of this dot dot dot action in order to do it, but see if you could write a general expression for year n.

Well, for year n, you're going to make that original thousand dollars at the beginning of year n. Then, you're going to have one thousand, one thousand times 1.05 for that thousand dollars that you deposit at the beginning of year n minus 1. This is just going to keep going, and it's going to go all the way to plus 1,000 dollars times 1.05 to the power of the number of years you've been compounding.

So, you could view this thousand dollars as the one that you put in year one. Then, how many years has it compounded? Well, when you go from one to two, you've compounded one year. When you go from one to three, you've compounded two years. So, when we're talking about the beginning of year n, you go up to the exponent that is one less than that, and so this is going to be to the n minus one power.

What we just did here is we've just constructed each one of these. When we're saying, "Okay, how much money do we have in our bank account at the beginning of year three?" or "How much do we have in our bank account at the beginning of year n?" These are geometric series, and I'll write that word down: geometric series.

Now, just as a little bit of a review, or it might not be review, it might be a primer, series are related to sequences, and you can really view series as sums of sequences.

Let me go down a little bit so that you can have a little bit more space. A sequence is an ordered list of numbers. A sequence might be something like, well, let's say an or we have a geometric sequence. In a geometric sequence, each successive term is the previous term times a fixed number. So, let's say we start at two, and every time we multiply by three.

So, we'll go from two, two times three is six, six times three is 18, 18 times three is 54. This is a geometric sequence: an ordered list of numbers. Now, if we want to think about the geometric series, or the one that's analogous to this, is that we would sum the terms here.

So this would be 2 plus 6 plus 18 plus 54. Or we could even write it, and this will look similar to what we had just done with our little savings example, as this is 2 plus 2 times three plus 2 times three squared plus 2 times three to the third power.

With the geometric series, you're going to have a sum where each successive term in the expression is equal to, if you put them all in order, is going to be equal to the term before it times a fixed amount. So, the second term is equal to the first term times three, and we're summing them in a sequence.

You're just looking at it; it's an ordered list, so to speak. But here, you are actually adding up the ordered list. So, what we just saw in this example is what a geometric series is, but also a famous example of how it's useful. This is just scratching the surface.

If you were to go further in finance or in business, you'll actually see geometric series popping up all over the place.