Exponential vs. linear growth over time

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 exede company a's payment so pause this video and try to work that out alright let's work this out together so let's let me set up a little bit of a table so this is going to be first column I'm going to have is month first column is month 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 we'll increase the amount each month by $5,000 so month two will be 5,000 more will get to $15,000 a month three will get two will get to $20,000 for will get to $25,000 five I think you get the point we'll get to $30,000 six will get to $35,000 seven will get to $40,000 let me scroll down a little bit month a they'll stop there month eight we will get to $45,000 and let me just extend these lines a little bit now let's think about was going to happen with Company B Company B is offering 500 for the first month it'll double their payment each month so the second month is going to be double that so that's going to be $1,000 then we're double that again $2,000 we're in double that again four thousand dollars double that again sixteen thousand dollars double that again thirty-two thousand dollars double that the Louvre I skipped one I went from four thousand two sixteen thousand four thousand eight thousand dollars then we doubled it again sixteen thousand dollars again thirty-two I sound like my two-year-old again thirty-two thousand dollars then we get to sixty-four thousand dollars and at that point something interesting happens actually good that I went to the eighth month because every month before the eighth month company age payment was higher until at eight month and that eight 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 five thousand plus five thousand it increases by five thousand 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 one 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 could 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 it's going to be described by a line so it could look something like this linear function is always going to be a line of some slope and an exponential function even though it might start a little bit slower it's eventually 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 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 of scoring two percent or three percent it still will eventually pass up the linear function which is pretty cool