Hey, everyone. So sometimes in some problems, you may be asked to write a general formula when you're given a recursive formula, like the problem we're going to work out down here in this example. We're going to use this to calculate the 15th term in this sequence. Remember, trying to do this just recursively is going to take forever. That's why these general formulas are really powerful. So let's take a look here. We already have a recursive formula for an arithmetic sequence, where the next term is just the previous term plus 3. We have the starting term, which is 2. That's what we need for an arithmetic sequence. So how do we actually get the general formula from this? It's actually really straightforward. Remember, all you need for a general formula is you just need the first number in the sequence, which is actually already given to us, and then you just need the common difference, lowercase d. And then you just multiply it by the index minus 1, n minus 1. That's all you need for the general formula. So if you're given the recursive formula for this, remember that this number that goes in front of the previous sequence or after the previous sequence, this is d. This is your common difference d. So we actually already have the two numbers that we need for our general formula, and we can sort of convert one formula into another. It's very straightforward. So if I can pull both of these things together into our formula here, what this says is that the nth term in the sequence is going to be the first term plus the common difference times n minus 1. Alright? So in other words, I'm just going to take that first number of the sequence, which I already know is 2, that's going to be 2, and I'm going to take the common difference, which I know is 3, and I'm going to multiply this by the index n minus 1. Alright? And that's going to be my nth term. This is the general formula for the sequence over here. In fact, you can use this to calculate the first two terms, and what you'll see here is that an2=5,an3=8,an4=11, just as we would get with a recursive, with a recursive formula. Alright? So that's the general formula. It's just the first term plus the common difference times n minus 1. Now let's use this to calculate the 15th term in the sequence. So the 15th term of the sequence is just where n equals 15, and you just plug that into your formula. The first term in the sequence is 2 + 3 times, and then the only place where you plug in the n is in the n minus 1. So this is going to be 15 minus 1. So in other words, this is just going to be 2+3×14, and 2+3×14 gives us a 15th term of 44. So that's the general formula, and the 15th term of the sequence is 44. Pretty straightforward. Alright? Thanks for watching. Let me know if you have any questions. Thanks. We'll see you in the next one.
Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
9. Sequences, Series, & Induction
Arithmetic Sequences
Video duration:
2mPlay a video:
Related Videos
Related Practice
Arithmetic Sequences practice set
![](/channels/images/assetPage/ctaCharacter.png)