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# Recursive formulas for arithmetic sequences

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

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• 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.
• 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.
• I find recursive sequences really, really confusing. Can anyone explain the basic rules for them? Thanks
• 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.
• what do those weird fractions mean? I've never seen that kind of writing and I'm confused
• What are recursive and explicit formulas?
• 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.
• 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?
• 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.
• What would 1.1, 1.9, 2.7, 3.5 be recursive formula
• 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.
• Why would we ever use a recursive formula instead of an explicit formula for any sequences, is it not more tedious and time consuming?
• 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.
• How do you figure out b when the g is out side of the equation?
• like when g(n-1) wouldn't the g(n) equal 4 and -1 and be 3 the anwser
• 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?
• 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.
g(n) = {A ` `if n=1