Find the sum of the arithmetic series root two, negative root two, negative three root two, all the way up to negative nine root two.
To work out this problem, we actually have a formula which helps us work out the sum of an arithmetic series. And that formula is the sum of 𝑛 numbers of terms is equal to the number of terms over two multiplied by 𝑎 one plus 𝑎 𝑛. And this is where 𝑛 — as we said — is the number of terms, 𝑎 one is our first term, and 𝑎 𝑛 is our last term.
Okay, now, we know what the formula is and actually know what each part is. Let’s use it to actually find the sum of our arithmetic series. But whenever I’m working with this type of problems, what I always do is actually write down what each part is first. So I am gonna start with 𝑎 one cause that’s our first term, which is just root two. And then, we’ve got our last term 𝑎 𝑛, which is negative nine root two. However, we don’t know what 𝑛 is because we actually don’t know how many terms there are in our series.
Well, in order to actually find out how many terms there are in our series, we have to think about what is an arithmetic series. Well, we’ve got a definition here. And that says an arithmetic series or sequence is a series where the difference between each term is constant. Okay, great, but how does this help us? Well, what we can do then is work out the common difference between our terms because this is gonna help us to actually determine how many terms there are in our series.
So to find 𝑑, I’m going to actually subtract the first term from the second term. I could have done the second term from the third term because either would've given us the same result because as we said there is a common difference. So we’ve got negative root two minus root two, which gives us negative two root two. Okay, great, so this is our common difference.
But still how does this help us work out how many terms there are? Well, what we can do is actually start our third term, which is negative three root two, and then use the common difference to see how many terms there are until we reach our last term, which is negative nine root two. Well, if we subtract two root two from negative three root two, we're gonna get negative five root two. So this is gonna be our fourth term.
So let’s do it again. Let’s subtract another two root two. Well, this gives us negative seven root two. Okay, this is our fifth term. And then what would happen if we subtract another two root two, we’d actually get to negative nine root two. So therefore, negative nine root two would be our sixth term. So therefore, we know that there’re actually gonna be six terms in our series.
Okay, brilliant, so now we’ve got 𝑛, we have 𝑎 one, and we have 𝑎 𝑛. We can actually substitute these into our formula to find the sum of our arithmetic series. So we have that the sum of the first six terms, which is our 𝑛, is equal to 𝑛 over two — so six over two — multiplied by our first term, which is root two, plus our last term which is negative nine root two which is gonna give us three multiplied by negative eight root two because root two plus negative nine root two gives us negative eight root two.
So therefore, we can say that the sum of the arithmetic series is negative 24 root two. And we got this because we multiplied three by negative eight, which gives us negative 24 then root two.