If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Finite geometric series formula

CCSS.Math:

## Video transcript

let's say that we are dealing with a geometric series and there are some things that we know about this geometric series for example we know that the first term of our geometric series is a so that is our first first term we also know the common ratio of our geometric series and we're gonna call that R so this is the common ratio and we also know that it's a finite geometric series so let me write this it is finite so there's a finite number of terms and let's say that n is equal to the number of terms the number of terms and we're gonna use a notation we're going to use the notation S sub n to donate to denote the sum the sum of first first of first n terms and the goal of this whole video is using this information coming up with a general formula for the sum of the first n terms to a formula for evaluating a geometric series so let's write out S sub n just just get a feeling for what it would look like so S sub n is going to be equal to where you have your first term here which is an A and then what's our second term going to be what's going to be this is a geometric series so it's going to be a times the common ratio so it's going to be the first term times the common ratio so the first term times R now what's the third term going to be well it's going to be the second term times our common ratio again so it's going to be a R times R or a R squared a R squared and we could go all the way to our nth term so we're going to go all the way to the nth term and you might be tempted to say it's going to be a times R to the nth power but we have to be careful here because notice our first term is really a r to the 0th power second term is a R to the first power our third term is a R to the second power so whatever term we're on the exponent is that that term number minus one so if we're on the nth term it's going to be a R to the N minus ones power so we want to come up with a nice clean formula for evaluating this and we're gonna instance we're gonna use a little trick to do it to do it we're gonna think about what R times the sum is and we're going to subtract that out so we're gonna take the R times that some R times times the sum of the first n terms actually let's just multiply negative R negative R times it's something that we can just add these two things and you'll see that it cleans this thing up nicely so what is this going to be equal to this is going to be equal to well if you multiply if we multiply a times negative R we will get we will get negative a R and I'm just gonna write it right underneath this one so if you multiply this times negative R I'm just gonna multiply every one of these terms by negative R that's equivalent of multiplying negative R times the sum I'm distributing the negative R so if I multiply it times this term a times negative R that's going to be negative that's going to be negative a R and if I multiply a R times negative R that's going to be negative a R squared negative a r squared you might see where this is going and just to be clear what's what's going on that's that term times negative R times negative R this is that term times negative R and we would keep going all the way to the the term the term this the term before this times negative R so if I the term before this times negative R is going to be a is going to be negative so you let me put subtraction signs it's going to be negative a times times R to the n minus 1 power R to the n minus 1 power that was a term right before this that was a times R to the n minus 2 times negative R is going to give us this so it's gonna get us right over there and then finally we take this last term and you multiply it by negative R what do you get you get negative a negative a and then times R to the N R to the N you multiply this times the negative U at the negative a and then R to the N minus one times R or times R to the first well this is going to be R to the N and now what's interesting here is this we can add up the left side and we can add up the right hand side so let's do that so let's do that on the left hand side we get S sub n minus R minus R times S sub n S sub N and on the right hand side we have something very cool happening notice it's a we still have that the a sits there but everything else except for this last thing is going to is going to cancel out so these two are gonna cancel out these two are gonna cancel out let me do that a little neater these two are gonna cancel out these two are going to be cancel out and all we're gonna have left with is a negative a R to the end so it's going to be a minus a times R to the nth power and now we can just solve for S sub N and we have our formula what we were looking for so let's see we can factor out an S sub n on the left hand side so you get an S sub n the sum of our first n terms you factor that out so it's going to be x times 1 minus R is going to be equal to and on the right hand side we can actually factor out an a so it's going to be a times 1 minus R to the N and so to solve for S sub n the sum of our first n terms we deserve a little bit of a drumroll here S sub n is going to be equal to this divided by 1 minus R so it's going to be a times 1 minus R to the N over 1 minus R and we're done we have figured out our our formula for the sum or for the sum of a finite for a finite geometric series and so in the next few videos or in future videos we will apply this and I encourage you whenever you whenever you use this formula it's very important now that you know where it came from that you really keep close track of how many terms you are actually summing up sometimes you might have a Sigma T Sigma notation it starts it might start its index at zero and then goes up to a number in which case you're gonna have that number plus one term so you have to be very careful this is the number of terms this is the first term here we define it up here and is the number of terms the first n terms are is our common ratio