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CCSS.Math:

a new social media site boasts that its user base has increased 47% each month for the past year the number of users on January 1st of last year was 50,000 50,000 which expression below gives a total number of new users in thousands that were added through month n of the past year where one is less than or equal to n which is less than or equal to 12 and they give some choices of expressions for the total number of new users that were added through month N and I encourage you to now pause this video and try to think about which of these expressions actually show that that gives that value well to tackle it I'm going to make a little bit of a table here so let's say we have month month and then we have users so starting users so users at the start and then let's think about the users added I want to put give myself some space to work with and then users at the end of the month users at the end of the month so in one month in or one which is January we can assume we started with 50,000 users they want us to write the expression in thousands so we started with 50,000 users and how many did we add well we added 47% of 50,000 so 50 times 47% so times 0.47 so how many do we end with well 50 plus 50 times 0.47 that's going to be 50 that's going to be 50 times one to the 15 green that's going to be 50 times one point four seven fifty times one point four seven if this isn't clear just think about this this you could rewrite as 50 times one so 50 times one plus 50 times 0.47 that's going to be 50 times one plus 0.47 one plus one plus is 0.47 or one point four seven so it's going to be this thing right over here so now let's go to month - let's go to a month - we start with what we ended the last month so fifty I could just copy and paste this actually just do that so that's what so copy and paste so that's what we start with now what are we going to add well we're going to add that this will we started with times 0.47 times times 0.47 and so what are we going to end with well if you sum these two and you could write this as you could write it this way this is going to be this is going to be this thing times times one point four seven times one point four seven or or we could just write this as 50 times one point four seven squared and you might start seeing a pattern now let's go to month three so month three what do we start the month with we start the month with this thing let me copy and paste this so copy and paste we start with that what do we add we're going to take that and we're going to multiply it times 47% we're going to multiply it times 0.47 and so what are we going to end up with we're going to have this times 1 plus this times 0.47 that is going to be equal to that that times times one point four seven one point four seven or we could just write this as 50 times one point four seven to the third power so what's the pattern here well in each month we're going to be starting with 50 times one point four seven to a power 1 less than the month in the third month the power here is two and the second month the power here is one in the first month the power here you don't see it but you could view this as x 1 point 4 7 to the 0 power so first month zero power second month first second month you have the first power third month you have the second power over here so if we're thinking about the nth month if we think about the nth month this is going to be a 50 times 1.47 1.47 to the n minus 1 power n minus 1 power is what we're going to start the month with now what are we going to add in the nth month what's going to be that times 47% so it's going to be just copy and paste that so let me just paste that so it's going to be that times 47% so x times 0.47 and then what are we going to end with when you add these two things you are going to get you are going to get 50 I'll just do it in the right color system copy paste it you're going to get 50 times one point four seven times one point four seven to the nth power to the nth power so let's think about how we can come up with the expression for the total number of new users and thousands that were added through through month n so there's a couple of ways to think about it you could say well how many total new you how many total users did we have at the end of month n well at the end of month end we had that many and then how many did we have at the beginning of the year well we hit 50,000 so how many total new users did we add through month n so we finished with this much so we finished with actually let me just write it down so we just let me just so we finished with that much and let me paste that so that's what we finished with and we started with 50,000 users we started with 50,000 users so this is essentially how many we added through month and now do any of our expressions look like that well no not quite if this one had a minus 50 right over here if that said minus 50 then that would have done the trick but this doesn't do it and none of the others really seem to either have this form or seem to be something that would be very easy to manipulate in this form so that's one way to do it but that's not one of our choices so what's another way of thinking about it well we could literally just add how many new users we added by month so we literally could just add all of these things right over here we could literally just add all of these things right over here and before I so let's see we could literally just add all of all of these terms but let me simplify it a little bit so what are some common factors that we see in all of these well we see they all have a 50 they all have a 50 and they all have something being multiplied by 0.47 so let's factor out a 0.47 and a 50 so let's factor that out so this is going to be equal to if we were to sum all of this up it's going to be 0.47 times 50 times 50 and then what's left over times so in the first month if you factor those two things out you're going to just left be left with a 1 in the second month if you factor out the 50 and the 0.47 you're left with one point four seven in the third month in the third month if you factor that and that out you're left with one point four seven squared and we're going to go all the way to the nth month if you factor out that and that you're left with one point four seven to the N minus one power so which of those expressions look like that well this is exactly this is exactly this second expression right over here this is exactly with what we came up with and we're done