now that you’ve tried to figure out this type of problem yourself, now we can discuss a method, using sequences and series that might really make this an easier problem. Using sequences and series, will even allow us to find the specific amounts of interest added after each compounding period and also allow us to find the total interest added to an investment after a certain amount of time. There are a couple equations you will be using for this project and will see throughout this chapter that we will go
over here. There are two types of sequences that you have encountered up to now: arithmetic and geometric. We will only be dealing with geometric sequences and series as our common ratio between terms is just going to be the interest rate of the investment. Let’s go through this a bit slower to explain how we will be using Sequences and Series to our advantage. As an example we put $100 into an investment with an annual interest rate of 5%. So we can describe the amounts we receive as interest in terms of a sequence of values from one year to the next. So in
the first year we will have 5% interest, or 5% of the initial investment added on, or 0.05*100 = 5. Thus $5 is the first year’s interest. Next we take that amount and multiply it by 100% plus 5%, because we are increasing the amount of interest we gain by 5%, not just taking 5% of the interest in the previous year. Remember this is only the interest we gain in the second year, not the total interest gained after
two years, there is a difference. So we take that interest from the year before, $5, and multiply it by 1.05, 1.05*5 = 5.25. Thus in the second year we gain $5.25 in interest. And so now we can see there is a pattern, multiplying the interest gained in the previous year by 1.05 to get the interest gained in the next year, and it makes a geometric sequence,
$5, $5.25, $5.51, $5.79, $6.08, etc.