Find the number of terms in the arithmetic sequence, whose first term is 11 and last term is 81, where the sum of all the terms is 506.
Resolving a problem like this, I first of all like to get down all the information that we have. Our first bit of information is that the first term is 11, so we can say that 𝑎 is equal to 11. As when we’re working with sequences, the terminology we use is 𝑎 is the first term. Our second piece of information is that the last term is 81, so we can say that 𝐿 is equal to 81 cause again the notation we use is cut to 𝐿 is equal to the last term. Finally, the last piece of information we have is the fact that the sum of all the terms is 506. So we can say 𝑆𝑛 is equal to 506. Again, cause of notation, that means the sum of 𝑛 terms.
Great! So that’s all the information that we have. And then finally, I’d like to make a note of what we’re actually looking for. So in this case, we’re looking for the number of terms. So we’ve got now that 𝑛 is something that we don’t know, so it’s what we need to find. So I put that down there. So now great, we know what we have, we know what we need to find, so let’s crack on and find out what 𝑛 is.
When we’re actually looking to find the sum of an arithmetic sequence, we can use formula to help us. Here are two formulas that we can have a look at. Well, first of all, we need to decide which of the formulas we can see here are we going to use. The way we can do that is by looking at the values we have and seeing which formula that’ll actually fit into.
Both formulae have 𝑎, so that can still allow us to use either of them. However, only our second formula has 𝐿, the last term, so therefore we know that that’s the formula that we’re going to be using for this problem. We can also see that we can’t use the first formula as it has a 𝑑, which is a common difference, and we don’t have the common difference and we can’t work out the common difference because we don’t actually know two consecutive terms. Great! So let’s get on and solve the problem.
So the first stage is to substitute our values in. First of all, we have the sum of all the terms which is 506, so that’s equal to 𝑛 the number of terms divided by two, and that’s the 𝑛 that we want to find. Then, it’s 11, which is our first term 𝑎, plus 81, which is our last term 𝐿. So we can now solve the equation to find 𝑛.
We start by multiplying both sides of the equation by two, and we’ve also added the 11 and 81 inside the parentheses. So we get 1012 is equal to 𝑛 multiplied by 92, which can be rewritten as 1012 equals 92𝑛. Then finally, we divide both sides by 92, which leaves us with 11 equals 𝑛 or 𝑛 equals 11. Therefore, we have solved the problem as we can say there are 11 terms in the sequence.