Formula for continuously compounding interest

Let's say that we're looking to borrow $50. So we can say that our principal is $50. We're going to borrow it for three years. So our time – let's say t – in years is 3. And let's say we're not going to just compound per year. We're going to compound four times a year\ – or every three months And let's say that our interest rate – if we were only compounding once per year – would be 10%. But since you're going to compound four times a year, what we're going to see in an expression is that we're going to divide this by 4 to see how much we compound each period. So 10% is the same thing as 0.10. So let's write an expression. And I encourage you actually to pause this video and try to write an expression for the amount you would have to pay back if you were to do this. If you were to borrow $50 over three years, compounding 4 times a year, each period, you would be compounding 10% divided by 4%. How much would you have to pay back in three years? Well let's write it out. $50, that's your principal. And you're going to multiply that – So you're going to compound it. each time, each period, each of these 3 × 4 periods. You have 3 years. Each of them are divided into 4 sections. So you're going to have 12 periods. So each of them, you're going to compound by 1 + r. We'll write that as a decimal: 0.10. – divided by the number of time that you're compounding per year – to the – (Well, you'd be raising it to the nth power if this was only over 1 year.) So there are 4 periods. And you'd raise it to the 4th power if it was only one year. But this is three years. So you're going to be doing this three – you're going to have 4 periods 3 times. Let me write this. So is going to be four – Actually, let me write this –