>> Here's a problem involving a finite geometric series. A deposit of $50 is made at the beginning of each month for five years in an account that pays five percent compounded monthly. What is the balance A in the account at the end of five years? Well, we know that our formula for figuring this sort of thing, that is, the, the amount in an account at the end of a certain period for a particular investment is this. P times 1 plus R over N to the NT power. Well, we're going to have to use this formula a number of times, aren't we? After all, each deposit is earning interest for a different period of time. In month one, we have a $50 deposit, which earns interest for the entire 60-month period. In month two, we have a $50 deposit which earns interest for 59 months, and we go all the way down to the last month when we make a $50 deposit, and it earns interest for only 1 month. Let's look at the situation algebraically. In month one, the deposit is $50. So that's the principal amount of investment. The rate is five percent, or five hundreds. N, the number of compoundings [phonetic] per year is 12. And T, the time is five years. Now, that time, T, is the item that changes from month to month, but for the first month, the time is the entire five-year period. Now notice. Five times 12 is 60. So our first investment is earning interest for 60 months. On the other hand, our investment in month two is earning interest for only 59 months, and then the next one would be 58 and so on down to month 60 in which our investment of $50 is earning interest for only one month. Well, notice, we have an interesting sequence here, and our accounting mechanism can be thought like this. Instead of writing this as, with our, our exponential accounting mechanism as 60, 59, 58, all the way down to 1, let's rewrite in a more logical order in the 1, 2, 3, 4. You see, the more logical counting order like this. Well, writing it like this, we see that we have the, the sum of a geometric sequence. And we can write that sequence like this. The summation as N runs from 1 to 60 of 50 times 1 plus five hundredths over 12 to the N power where N, now, is our counting mechanism. Our formula for figuring the, the sum in this situation is this. Now, ace of 1 is the first item in the sequence, and it is found by replacing N with 1. So A then becomes simply 50 times 1 plus five hundredths over 12. Now, in the bracket, the R value is simply the base of our counting mechanism. So in this case R is 1 plus five hundredths over 12. Now notice the N. Now the N involved here is in the formula is the number of items we're counting, and in the equation above, the N stood for the counting mechanisms. Now this seems to be confusing, but I've set it up so that both of these are lettered N just to point out the difference between the two and the possible confusion that can occur. The N in the A formula is simply a counting mechanism, and the N involved in our formula for figuring the sum of the geometric sequence is the number of items being counted, and that number of items here, of course, is 60. OK. So we initially replaced with 60 in our formula. Now the R down below is the same as the R above. It's the base of the counting mechanism. One plus five hundredths over 12. Now, we simply go through and perform the indicated operations by punching our calculator a bit, and we find A to be approximately 3,414 and 47 hundredths, which we interrupt in dollars and cents, and write our answer. The balance in the account at the end of five years is $3,414.47.