Question Video: Writing the Recursive Formula of an Arithmetic Sequence Mathematics

Write down a recursive formula for the sequence 5, 7, 9, 11, 13, ….

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Video Transcript

Write down a recursive formula for the sequence five, seven, nine, 11, 13, and so on.

Well, a good place to start is to work out what’s the difference between each term. To get from five to seven, I need to add two. From seven to nine, add two. And to get between the next two terms, I add two. And the next two terms again, I add two again. And it’s a good idea to try to describe what’s going on before you try to put it into a formula. So, to get from one term to the next, I’m adding two every time. And in our formula then, if I take a term, let’s call it 𝑎 𝑛, and I add two to that term, it’s going to give me the next term, 𝑎 𝑛 plus one.

Well, actually, its sort of as easy as that. That’s the basic formula. But we do need to tell it where to start and we need to define the values of 𝑛 that are gonna work in this formula. Well, our first term was five, so 𝑎 one is five. And we want to use values of 𝑛 that generate terms 𝑎 one, 𝑎 two, 𝑎 three, 𝑎 four, 𝑎 five, and so on. Now, in our formula, we’ve got 𝑎 𝑛, and then we’ve got 𝑎 𝑛 plus one. So, we can have 𝑛 has all the values from one upwards. So, that’s our formula. 𝑎 𝑛 plus one equals 𝑎 𝑛 plus two, where 𝑎 one is five, and 𝑛 is greater than or equal to one and an integer.

Now, I could’ve formulated that slightly differently. So, I could’ve said the 𝑛th term, 𝑎 𝑛, is simply equal to the previous term 𝑎 𝑛 minus one plus two. And again, the first term 𝑎 one is equal to five. But let’s think about the values of 𝑛. If I put 𝑛 equal to one, I’d be saying 𝑎 one, the first term, is equal to 𝑎 one minus one, 𝑎 zero. I’d be talking about the zeroth term plus two. And I don’t have a zeroth term, so I’m gonna define 𝑛 from two onwards. That gives me an alternative recursive formula.

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