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### Course: Operations and Algebraic Thinking 229+ > Unit 7

Lesson 2: Constructing arithmetic sequences- Recursive formulas for arithmetic sequences
- Recursive formulas for arithmetic sequences
- Recursive formulas for arithmetic sequences
- Explicit formulas for arithmetic sequences
- Explicit formulas for arithmetic sequences
- Explicit formulas for arithmetic sequences
- Arithmetic sequence problem
- Converting recursive & explicit forms of arithmetic sequences
- Converting recursive & explicit forms of arithmetic sequences
- Converting recursive & explicit forms of arithmetic sequences
- Arithmetic sequences review

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# Recursive formulas for arithmetic sequences

Sal finds the recursive formula of the arithmetic sequence 4, 3⅘, 3⅗, 3⅖,...

## Want to join the conversation?

- Why are recursive formulas ever used? It seems like explicit formulas would always be better, as they are so much easier to apply to larger numbers.(56 votes)
- Recursive formulas are a different way to solve a problem. Recursion is a thought process that uses previous data in a step-by-step manner. Recursion is used extensively in computer science.(55 votes)

- I find recursive sequences really, really confusing. Can anyone explain the basic rules for them? Thanks(15 votes)
- You're given two things:

1. The first term of the sequence

2. A rule for how to find the next term of a sequence given a previous term

For example, we could define a sequence this way:

The first term is 1.

Each term is double the previous term.

So to find the second term, we take the term before it (1) and double it. So the second term is 2.

Now that we have the second term, we can double it to find the third term: 4

We can double the third term to get the fourth: 8

And so on.(27 votes)

- what do those weird fractions mean? I've never seen that kind of writing and I'm confused(6 votes)
- Those are Mixed numbers. Look for them inside the fractions in Khan Academy.

Here's a Link: https://www.khanacademy.org/math/pre-algebra/pre-algebra-fractions/pre-algebra-mixed-number/v/changing-a-mixed-number-to-an-improper-fraction(16 votes)

- What are recursive and explicit formulas?(6 votes)
- A recursive formula always uses the preceding term to define the next term of the sequence. Sequences can have the same formula but because they start with a different number, they are different patterns.(3 votes)

- How do you solve an arithmetic recursive equation where you don't have the value of n, just the common difference and the first term?(3 votes)
- You're making it more complicated than it is.

"n" is the term you are looking for. If you want the 2nd term, then n=2; for 3rd term n=3; etc.

The recursive equation for an arithmetic squence is:

f(1) = the value for the 1st term.

f(n) = f(n-1) + common difference.

For example: if 1st term = 5 and common difference is 3, your equation becomes:

f(1) = 5

f(n) = f(n-1)+3

Hope this helps.(6 votes)

- What would 1.1, 1.9, 2.7, 3.5 be recursive formula(4 votes)
- g(n) ={1.1 if n = 1}

{g(n-1) + 0.8 if n > 1}

You take the first number (1.1), and put that as the (if n = 1).

Then you take the n-1th term (if n = 2, [n-1] = 1), and add the difference of two*consecutive*terms. For anyone who doesn't know, consecutive means one after the other.(3 votes)

- Why would we ever use a recursive formula instead of an explicit formula for any sequences, is it not more tedious and time consuming?(4 votes)
- Great question!

Recursive formula is very tedious, but sometimes it works a little easier. If you are trying to find the fourth or third term, you can use recursive form. But if you are trying to find the 41th term, the explicit formula is easier.(2 votes)

- Could someone please tell me when we're ever going to need to know this?(3 votes)
- One of google's reasons for using recursive sequences is sorting documents. I also thing they are used in programing.(4 votes)

- How do you figure out b when the g is out side of the equation?(3 votes)
- like when g(n-1) wouldn't the g(n) equal 4 and -1 and be 3 the anwser(2 votes)

- Would it be correct to say that, using the recursive formula, we can't work out the 1st term because, we are given the first term, we can only use it to work out the terms greater than 1?(2 votes)
- Yes, but it also means that if we are given any term (say the 20), then we could find the next one, it does not have to start at 1.(3 votes)

## Video transcript

- [Voiceover] g is a function that describes an arithmetic sequence. Here are the first few
terms of the sequence. So let's say the first term is four, second term is 3 4/5, third term is 3 3/5, fourth term is 3 2/5. Find the values of the
missing parameters A and B in the following recursive
definition of the sequence. So they say the nth term is going to be equal to A if n is equal to one and it's going to be
equal to g of n minus one plus B if n is greater than one. And so I encourage you to pause this video and see if you could figure out what A and B are going to be. Well, the first one to figure out, A is actually pretty straightforward. If n is equal to one, if n is equal to one, the first term when n equals one is four. So A is equal to four. So we could write this as g of n is equal to four if n is equal to one. And now let's think about the second line. The second line is interesting. It's saying it's going to be
equal to the previous term, g of n minus one. This means the n minus oneth term, plus B, will give you the nth term. Let's just think about what's happening with this arithmetic sequence. When I go from the first
term to the second term, what have I done? Looks like I have subtracted 1/5, so minus 1/5, and then it's an arithmetic sequence so I should subtract or add
the same amount to every time, and I am, I'm subtracting 1/5, and so I am subtracting 1/5. And so one way to think about it, if we were to go the other way, we could say, for example, that g of four is equal to g of three minus 1/5, minus 1/5. You see that right over here. g of three is this. You subtract 1/5, you get g of four. You see that right over there and of course I could have written this like g of four is equal to g of four minus one minus 1/5. So if you look at this way, you could see that if I'm
trying to find the nth term, it's gonna be the n minus oneth term plus negative 1/5, so B is negative 1/5. Once again, if I'm trying
to find the fourth term, if n is equal to four, I'm not gonna use this first case 'cause this has to be for n equals one, so if n equals four, I
would use the second case, so then it would be g of four minus one, it would be g of three minus 1/5. And so we could say g of n is equal to g of n minus one, so the term right before that minus 1/5 if n is greater than one. But for the sake of this problem, we see that A is equal to four and B is equal to negative 1/5.