# Finite geometric series word problem: social media

## Video udskrift

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. Which expression below
gives the total number of new users in thousands
that were added through month n of the past year, where 1 is
less than or equal to n, which is less than or equal to 12. And they give us 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, and
then we have starting users-- so users at the
start-- and then let's think about the users added. I want to give myself
some space to work with. And then users at
the end of the month. So in month 1, which is
January we can assume, we started with 50,000 users. They want us to write
the expression in 1,000. 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 times-- I'm gonna
do the 50 in green-- that's going to be 50 times 1.47. If this isn't clear,
just think about this. This you could
rewrite as 50 times 1. So 50 times 1 plus
50 times 0.47, that's going to be
50 times 1 plus 0.47. Or 1.47. So it's going to be this
thing right over here. So now let's go to month 2. We start with what we
ended the last month. So I could just copy
and paste this actually. Let me just do that. So copy and paste. So that's what we start with. Now what are we going to add? Well we're going to add
this, what we started with, times 0.47. And so what are we
going to end with? Well if you sum these two-- and
you could write it this way-- this is going to be
this thing times 1.47. Or we could just write this
is 50 times 1.47 squared. And you might start
seeing a pattern now. Let's go to month 3. So month 3: 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? Well 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 gonna have this times
1 plus this times 0.47. That is going to be equal
to that times 1.471. Or we could just write this as
50 times 1.47 to the 3rd power. So what's the pattern here? Well in each month, we're
going to be starting with 50 times 1.47 to a
power 1 less than the month. In the third month,
the power here is 2. In the second month,
the power here is 1. In the first month, the
power here-- you don't see it but you can view this as
times 1.47 to the 0 power. So 1st month, 0 power. Third month, you
have the 1st power. Third month, you have the
second power over here. So if we're thinking
about the nth month this is going to
be 50 times 1.47 to the n minus 1th power, is what we're
going to start the month with. Now what are we going
to add in the nth month? Well it's going to
be that times 47%. So it's going to be--
we'll just copy and paste that-- so it's going
to be that times 47%. Times 0.47. And then, what are
we going to end with? Well when you add
these two things, you are going to get
50-- I'll just do it in the right colors instead
of copying and pasting it-- you're going to get 50
times 1.47 to the nth power. So let's think about
how we can come up with the expression for the
total number of new users in thousands that were
added through month n. So there's a couple of
ways to think about it. You could say, well how
many total new users did we have at the
end of month n? Well at the end of month
n, we had that many. And then how many did we have
at the beginning of the year? Well we have 50,000. So how many total new users
did we add through month n. So we finished with this much. Let me just write it down. So we just finished
with that much. And let me paste that. So that's what we finished with. And we started
with 50,000 users. So this is essentially how
many we added through month n. 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've 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 month by month. So we literally could just
add all of these things right over here. So let's see, we could literally
just add 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. 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 this up-- it's
going to be 0.47 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 be left with a 1. In the second month if you
factor out the 50 and the 0.47, you're left with 1.47. In the third month, if you
factor that and that out, you're left with 1.47 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 1.47 to the
n minus 1th power. So which of those
expressions look like that? Well this is exactly the second
expression right over here. This is exactly what
we came up with. And we're done.