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### Course: Algebra 2>Unit 3

Lesson 7: Geometric series

# Geometric series introduction

Learn how money grows in a bank account with geometric series! Discover how each deposit grows by a fixed percentage every year, creating a pattern. This pattern forms a geometric series, a useful concept in finance and business. Keep depositing and watch your money multiply!

## Want to join the conversation?

• I still cannot get it. Why do we get 1000 + 1000(1.05) which is 1000 + 1050 = 2050 on the first year; I think we should get 1000 + 1000(0.05) or 1000(1.05).

I would appreciate any help, thank You all for this project by the way.
• Every year he is depositing \$1000. However, the \$1000 deposited in previous years is still earning interest.
For example, in the first year, he deposits \$1000.
In the second year, he gains 5% interest on the \$1000, and now it becomes \$1050. But since he is depositing \$1000 every year, in the second year he has a total of \$2050, since he has not yet earned interest on the second \$1000 that he deposited. In the third year the first \$1000 that he deposited that had turned into \$1050, gains %5 interest and is again multiplied by 1.05. basically, the first hundred now has been multiplied by 1.05 twice. 1000(1.05)(1.05) or 1000(1.05)^2. The second \$1000 he deposited gains %5 interest and turns into 1000(1.05)^1 and then since it is the beginning of the year he deposits one more \$1000, which has not gained interest yet, so is 1000(1.05)^0 or just 1000.
So the pattern in the third year is:
1000 + 1000(1.05)^1 + 1000(1.05)^2 + 1000(1.05)^2.
Note that the highest exponent is 2 which is one less than the year number. So Khan is correct. You forgot that every year he deposits \$1000 and the money from the previous years collect more and more interest as the years go by.
• In the description above it reads: "For example, 1, 2, 4, 8,... is a geometric sequence"...

I understand the process of the sequence but I'm more looking for the history of the terminology so to say. Geometry is about (measuring) shapes and sizes. Why is this multiplying sequence not called a 'multiplying sequence', or a mathimatical sequence or what have you. Of all the possible branches of maths it had to be geometry. Why?
• It kinda just happened that way...

The ancient Greek mathematician Euclid first wrote about these types of sequences in his book Elements. Because so much of Euclid's Elements deals with geometry, these sequences ended up being called geometric sequences (even though they aren't technically geometric).

So the "geometric" label is an historical accident, but kind of interesting if you know the story.

Hope this helps!
• Why is this geometric series stuff part of the Factoring Polynomials unit?
• agreed lol, I think it's fascinating but a very jarring transition
• Multiplying the balance by the 1.05 is not 5% growth per year, it is 105% growth. I think it must be:
1000 + 1000(0.05) or 1000(1.05)
• 1.05 is just 5 percent growth plus the original if you just wanted the growth value you would do balance times .05 . That would be just growth
• I've been following the Algebra 2 series, and this section on geometric series is nested under Polynomial Factorization. Why is that? I didn't need to factor any polynomials to solve geometric series problems...
• It's still dealing with polynomials like ^n-1 which can go up to any number. So it still deals with polynomials. Also, you can also factorise these.(sometimes)
• So a question about sequences will say " What's the deposit value of the of the 8th day? " but about series would say " How much money does they have on the 8th day ? " Am I right ?
• Yes, that does seem correct. One asks for a specific term in the sequence, while the other asks for the sum of the terms.
• If the geometric series is 1000+(1000*1.05) to the n th power then wouldn't you get 2005? Why isn't it 1000+(1000*0.05).
• I think Sal meant that in addition to the 5% increase each year, we're also going to deposit an additional 1000 each year. So for each year, we add 1000 (the additional 1000) and we also times 1000 by 1.05 because we're keeping the 1000 in their and we have 5% interest.
I hope I wrote that clearly
• Would 1000+(1000(1.05)^n-1)^n work?
• I don't think the ^n at the end would work because you want to add each year's deposit including their accrued interest plus the 1000, not multiplying by itself. I don't know what the formula would be though, he never finished teaching it.
• So what was the actual formula for his little \$1000 per year example?? Was it 1000(1.05)^n-1 ? Coz he didn't make that very clear -_-
• Yes, you're right. To get the nth term in the geometric sequence, you would evaluate 1000(1.05)^(n-1). This is because we start with \$1000, and increase it by 5% every year. The minus 1 is added because technically, the 1st term is our starting point without any interest, so we need to offset the term number by 1 to get the actual number for the formula.
• What is the difference between a geometric sequence and an exponential function? They seem to be the same thing, so I am confused.
• A sequence is specifically gotten from a list of numbers, but can be transformed into an exponential function. Also since geometric sequences always have the same starting point there will be no horizontal or vertical shifts in the function.

## Video transcript

- [Instructor] In this video we're gonna study geometric series, and to understand that I'm gonna construct a little bit of a table to understand how our money could grow if we keep depositing, let's say, \$1,000 a year in a bank account. So, let's say this is the year, and we're gonna think about how much we have at the beginning of the year, and then this is the dollars in our account. And let's say that the bank is always willing to give us 5% per year, which is pretty good. It's very hard to find a bank account that will actually give you 5% growth per year. So that means if you put \$100 in at the end of a year, or exactly a year later it'd be \$105. If you put \$1,000 in, a year later, it'd be 1,050. It'd be 5% larger. And so, let's say that we want to put \$1,000 in per year, and 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 in the account. That's pretty straightforward. But then what happens in year two? I'm going to deposit \$1,000, but then that original \$1,000 that I have would have grown. So, I'm going to deposit \$1,000, and then the original \$1,000 that I put at the beginning of year one, that has now grown by 5%. Growing by 5% is the same thing as multiplying by 1.05. So, this is now going to be plus \$1,000 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, that year three deposit? Pause this video. 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 \$1,000 deposit, but now the money from year two has grown by 5%, so this is now going to be \$1,000 times 1.05. And then that money that we originally deposited from year one, that was 1,000 times 1.05 in year two, that's going to grow by another 5%. And so, this is going to be plus 1,000 times 1.05 times 1.05. We're growing by another 5%. 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, see if you can write a general expression. You're gonna 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 \$1,000 at the beginning of year n, and then you're going to have 1,000, 1,000, times 1.05 for that \$1,000 that you deposit at the beginning of year n minus one. And then this is just going to keep going, and it's going to go all the way to plus \$1,000 to times 1.05 to the power of the number of years you've been compounding. So, you could do this \$1,000 as the one that you put in year one, and 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. So, 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 will write that word down. Geo, 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. Sequences, and let me go down a little bit so that you can, so we have a little bit more space, a sequence is an ordered list of numbers. A sequence might be something like, well, let's say we have a geometric sequence, and 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, 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 two plus six plus 18 plus 54. Or we could even write it, and this would look similar to what we had just done with our little savings example is this is two plus two times three, plus two times three squared, plus two times three to the third power. And so, with the geometric series, you're going to have a sum where each successive term in the expression is equal to, if you put 'em 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 one, what a geometric series is, but also, a famous example of how it's useful, and 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.