Exponential vs. linear growth over time

- [Instructor] Company A is offering $10,000 for the first month and will increase the amount each month by $5,000. Company B is offering $500 for the first month and will double their payment each month. For which monthly payment will Company B's payment first exceed Company A's payment? So pause this video and try to work that out. All right, let's work this out together. So let me set up a little bit of a table. So this is going to be, the first column I'm gonna have is month. The first column is month. The second column is how much Company A is going to pay. And then third column, let's think about how much Company B is going to pay. Well, they tell us a few things. They say Company A is offering $10,000 for the first month. So in month one, Company A is offering $10,000, we'll assume, well, I'll just write the dollars there. And then Company B is offering $500 for the first month. $500 for the first month. But then they tell us Company A is offering, will increase the amount, each month by $5,000. So month two will be 5,000 more, we'll get to $15,000. Month three we'll get to $20,000. Four, we'll get to $25,000. Five, I think you get the point, we'll get to $30,000. Six, we'll get to $35,000. Seven, we'll get to $40,000. Let me scroll down a little bit. Month eight, I'll stop there. Month eight, we will get to $45,000. Let me extend these lines a little bit. Now let's think about what's gonna happen with Company B. Company B is offering 500 for the first month, but will double their payment each month. So the second month is gonna be double that. So that's going to be $1,000. Then we're gonna double that again, $2,000. Then we're gonna double that again, $4,000. Double that again, $16,000. Double that again, $32,000. Double that, oh, I skipped one. I went from 4,000 to 16,000. 4,000, $8,000. Then we double it again, $16,000. Again, 32, I sound like my two year old again. All right, $32,000. Then we get to $64,000. And at that point something interesting happens. It's actually good that I went to the eighth month because every month before the eighth month, Company A's payment was higher until that eighth month. In that eighth month, Company B is going to pay more. So first we can just answer their question. For which monthly payment will Company B's payment first exceed Company A's payment? Well, that is month eight, month eight. And there's a broader lesson going on here. You might recognize that the rate at which Company A's payment is increasing is linear. Every month it increases by the same amount. So plus 5,000, plus 5,000. It increases by 5,000 the same amount. Company B is increasing exponentially. It's increasing by the same factor every time, so we're multiplying by the same value every time. We're multiplying by two, we're multiplying by two, multiplying by two. And so there's actually a very interesting thing here that you can actually make the general statement that an exponential function will, we'll want something that is exponentially increasing, will eventually always surpass something that is linearly increasing. And it doesn't matter what the initial situation is, and it also doesn't even matter that rate of exponential increase, it will eventually always pass up something that's increasing linearly. And you can think about that visually if you like. If I were to draw a visual function, a linear function, so this is x-axis, this is the y-axis, a linear function, well, this is gonna be described by a line. So it could look something like this. A linear function is always gonna be a line of some slope. And an exponential function, even though it might start a little bit slower, it's eventually going to pass up the linear function. And this is going to be the case even if the linear function has a pretty high slope or a pretty high starting point, if it's something like that, and even if the exponential function is starting pretty slow, it will eventually, and even if it's compounding or growing relatively slow but exponentially, you know if it's going 2% or 3%, it still will eventually pass up the linear function which is pretty cool.