And so, with the geometric series, you're going to have a sum
#25M SERIES MAIN SEQUENCE PLUS#
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. Or we could even write it,Īnd this would look similar to what we had just done with Now, if we want to thinkĪbout the geometric series, or the one that's analogous to this, is that we would sum the terms here. Two times three is six, six times three is 18,ġ8 times three is 54. So, let's say we start at two, and every time we multiply by three. A sequence might be something like, well, let's say we haveĪ geometric sequence, and a geometric sequence, each successive term is the previous term Sequences, and let me go down a little bit so that you can, so we haveĪ little bit more space, a sequence is an ordered list of numbers. It might be a primer, series are related to sequences, and you can really view Now, just as a little bit of a review, or it might not be review,
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. Is we've just constructed each one of these when we're saying, okay, how much money do we The beginning of year n, you go up to the exponent 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. The way to plus $1,000 to times 1.05 to the power of the number of years And then this is just going to keep going, and it's going to go all To make that original $1,000 at the beginning of year n,Īnd 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. You're gonna have to do a little bit of this dot, dot, dotĪction in order to do it. So, do you see a general pattern that's going to happen here? Well, as you go to year n, inįact, pause the video again, see if you can write a general expression. Well, we could just rewrite this part right over here as 1.05 squared. And so, this is going to be plus 1,000 times 1.05 times 1.05. We originally deposited from year one, that wasġ,000 times 1.05 in year two, that's going to grow by another 5%.
Year two has grown by 5%, so this is now going toīe $1,000 times 1.05. Well, just like at theīeginning of year two and the beginning of year one, we're going to make $1,000 deposit, but now the money from
Now, what about theīeginning of year three? How much would I have in the bank account right when I've made thatįirst, that year three deposit? Pause this video. So, this is now going to be plus $1,000 times 1.05. So, I'm going to deposit $1,000, and then the original $1,000 that I put at the beginning of year one, But then what happens in year two? I'm going to deposit $1,000,īut then that original $1,000 that I have would have grown. The beginning of the year, I put in $1,000 in the account. Year three, and then see if we can come up withĪ general expression for the beginning of year n. 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,Īt the beginning of year two, at the beginning of $100 in at the end of a year, or exactly a year later it'd be $105. It's very hard to find a bank account that will actually give Is always willing to give us 5% per year, which is pretty good. So, let's say this is the year, and we're gonna thinkĪbout how much we have at the beginning of the year, and then this is theĭollars in our account. We keep depositing, let's say, $1,000 a year in a bank account. Understand that I'm gonna construct a little bit of a table to understand how our money could grow if Video we're gonna study geometric series, and to 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. Note that the highest exponent is 2 which is one less than the year number. basically, the first hundred now has been multiplied by 1.05 twice. 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. 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 second year, he gains 5% interest on the $1000, and now it becomes $1050. However, the $1000 deposited in previous years is still earning interest.įor example, in the first year, he deposits $1000.
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