 So, simple interest has several advantages. However, the effective rate drops as time passes. And while this might not seem to be a problem, it makes the mathematics more complicated. So we'll make the mathematics easier by considering a more complicated scenario. So from our formula for the effective interest rate over the nth time period, we can rearrange if we multiply both sides by a of n minus 1 we get, and we can solve for a of n. Now if we want the effective interest rate to be the same for all n, we can call it i and get. This formula a of n minus 1 times 1 plus i is an example of what mathematicians call a recurrence relation. It expresses the value of a quantity a n in terms of the value of a previous quantity a of n minus 1. It's often easy to write recurrence relations, we just did, but it's hard to work with them. For example, if we wanted to know a of 500, we'd have our recurrence relation a of 500 is a of 499 times 1 plus i. But now we need to know a of 499. Well, we can find that by using our recurrence relation because a of 499 is a of 498 times 1 plus i. But now we need to find a of 498, so this requires finding a of 498, which is a of 497 times 1 plus i, and so on. And so we say that we want a closed form expression for a of n. In other words, some way we can compute a of n directly without having to find all of the preceding values. So let's start by finding a of 1. Our recurrence relation says that it's equal to a of 0 times 1 plus i, but remember we'll always assume a of 0 is equal to 1. So a of 1 will be 1 plus i. Now let's consider a of 2. Our recurrence relation says that it's a of 1 times 1 plus i, but we've already figured out what a of 1 is. So a of 2 will be 1 plus i squared. A of 3, well, that's a of 2 times 1 plus i. We know what a of 2 is, 1 plus i squared, and so a of 3 is 1 plus i cubed, and a pattern begins to emerge. And so now we'll lather, rinse, repeat, and generalize to find a of n equals 1 plus i to the nth power. And this leads to the compound interest accumulation function. So the compound interest accumulation function for interest rate i is 1 plus i raised to the tth power. So remember we'll be assuming that our amount function akt is just going to be the amount invested k times the accumulation function a of t. And so the compound interest formula relates four quantities, the final amount akt, the principal k, the time t, and the interest rate i. And given any 3, we can solve for the fourth. So I suppose 500 is deposited for 8 years at compound interest with an annual rate of 3%, and let's compare the final amount to the same amount deposited for the same time at simple interest. We'll use the accumulation function, which tells us how much a dollar would become, then scale it by the $500 investment. We have t equal to 8, i the interest rate 3% converted to a decimal, and so we find a of 8, and so 500 would become 500 times 1.03 to the 8th, or about $633.39. In comparison, the accumulation function for simple interest would give us, so $500 would become $620. We could also try to find an interest rate, so let's find the interest rate required for a deposit of $200 to grow to $10,000 over 30 years. So as a general rule, unless otherwise specified, we'll assume that we're dealing with compound interest. We rarely work with simple interest outside of a few basic examples. So our amount function will be, and we know the amount we want, $10,000, and the starting principle, $200. We also know the time, 30 years. So the last thing we do is the first thing we take care of, so over on the right hand side we're multiplying by 200, so we'll divide by 200. The right hand side is a 30th power, so we'll take the 30th root, and then we'll subtract 1 to get the interest rate. We can also try to find an unknown time. Suppose $1,000 is deposited in an account bearing 12% interest compounded annually. How long before this amount triples in value? So we have our amount function, where we know the interest rate, 12%, and the principal balance, 1,000. And since we want the amount to triple, the final amount is going to be 3 times 1,000 or 3,000. And so now we can solve for t. We'll divide both sides by 1,000. And since this is an exponential equation, we'll hit both sides with the log and solve. And we find that t is approximately 9.69 years. Or suppose we want to retire in five years with $500,000. According to this email I just got, a Russian prince offers me a return of 45% per year. So how much should I send the Russian prince? We have afk of 5 is $500,000, that's the amount after five years. The interest rate is 45%, and the time is 5. And so we have... And since the right hand side is k multiplied by a quantity, we can divide both sides by that quantity to get k. And that gives us our value for k. And so I should give this Russian prince... Well, probably nothing. If an offer sounds too good to be true, it probably is.