Main content

## Get ready for Precalculus

### Unit 6: Lesson 1

Intro to sequences- Sequences intro
- Intro to arithmetic sequences
- Intro to arithmetic sequences
- Using arithmetic sequences formulas
- Intro to arithmetic sequence formulas
- Worked example: using recursive formula for arithmetic sequence
- Use arithmetic sequence formulas
- Intro to geometric sequences
- Using explicit formulas of geometric sequences
- Using recursive formulas of geometric sequences
- Use geometric sequence formulas

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Worked example: using recursive formula for arithmetic sequence

CCSS.Math: ,

Example finding the 4th term in a recursively defined arithmetic sequence.

## Want to join the conversation?

- when you put b(1) you get=12, in the data b(1)=-7, I didn't get it?(10 votes)
- Your interpreting this incorrectly.

Function "b" is a piecewise function. The top is telling you the pieces. If you need b(1), it = -7. For any other value of function "b", it is calculated using the 2nd row of the function's definition.

It appears that you used the 2nd row for all values, not just values of n>1. And, you assumed that b(1-1) = b(0) = 0. You can't make that assumption. This is why the 1st row of the function definition exists to tell you the starting value of the function.

Hope this helps.(11 votes)

- Is there a faster way to find terms using the recursive formula?(5 votes)
- you can convert a recursive to explicit very easily(5 votes)

- I was doing my homework and came across this question: "Calculate the second term of the recursive function in which f(1)= 3 and f(x)= 3f(x-1)." If anyone knows how to solve this please let me know!(2 votes)
- f(1) stands for the first term. I can see that the first term is 3. (3)f(x-1) is the recursive formula for a given geometric sequence. If we had 3+f(x-1), we would have an arithmetic sequence. Notice the 3 I put in parentheses. This is the common ratio. You must multiply that to the previous term to get the next term, since this is a geometric sequence. Since you need to find the second term, you simply must multiply the first term by 3. 3*3=9, so 9 is your second term.(5 votes)

- Couldn't I just identify that the first number of the sequence is -7 and that the reoccurring arithmetic operation in the sequence is add 12? Just by looking at the information that was given to me I could set up the equation -7 + 12 x 3 = 29. Wouldn't this be faster and easier?(3 votes)
- Is there a quicker way to solve this type of problem?(2 votes)
- you can convert the recursive formula to an explicit formula(3 votes)

- At1:52, Sal said that we d'ont know b(1) but it's mentioned in the question. So why did he say that ? do you think he didn't read it well ?(2 votes)
- Maybe when after that he said "..lets figure it out" he just meant that we don't b(1) just yet, but we have to read the question and realize that they have already given the answer to us.

Hope that made sense:)(1 vote)

- What happens if they ask us for the hundredth term? Do we have to recurse all the way back to the first term? We had a problem like this on a test that I got wrong.(2 votes)
- Couldn't I just identify that the first number of the sequence is -7 and that the reoccurring arithmetic operation in the sequence is add 12? Just by looking at the information that was given to me I could set up the equation -7 + 12 x 3 = 29. Wouldn't this be faster and easier?(1 vote)
- You basically changed the recursive definition of the sequence into its explicit definition. If you were asked to find the 4th term using the recursive version, then you can't use your technique. Otherwise, you method is fine.

Note: The whole purpose of this video was to show how to use the recursive version which is why Sal takes the steps that he did.(2 votes)

- Would you do the same if the sequence was decreasing?(1 vote)
- Yes you just end up with a negative sign(2 votes)

- I'm definitely misunderstanding something but why does it not work if you put -7 in for b, and 4 in for n.(1 vote)
- b(1) = -7

This means the first term = -7

This is a recursive formula. So, you need to know the prior term to calculate the next term.

If you need n=4 (the 4th term), then you must know the value of the 3rd term. Since -7 is the value of the 1st term, it won't work. As you see in the video, to find the 4th term, Sal needs to calculate b(2) - the 2nd term. Then b(3) - the 3rd term. And, then he can find b(4) - the 4th term.

Hope this helps.(2 votes)

## Video transcript

- [Instructor] We are
told b of one is equal to negative seven, and b of n is equal to b of n minus one plus
12, and they're asking us to find the fourth term in the sequence. So what we have up here, which you could use a function definition, it's really defining
the terms of a sequence. Especially if are to input
whole numbers in here, it's the index on your sequence. What we really wanna do
is, we wanna figure out what is b of four going to be equal to? Well if we just blindly apply this, we would say all right b of four, so be of n, is equal to
b of n minus one plus 12, so it's gonna be b of
four minus one plus 12. Well four minus one is just three, so it's going to be equal
to b of three plus 12. All I did is said okay, well
we're not trying to figure out, or we're not immediately
trying to figure out what b of one is. We're trying to figure
out what b of four is, so n is equal to four, so b
of four is going to be equal to b of four minus one,
or b of three, plus 12. To evaluate this, we have to
figure out what b of three is, so let's write that down. That's what's fun about
a recursive definition. You have to keep recursing
backwards, so b of three. Well if n is three, that's
going to be equal to b of, now n minus one is two, b of two plus 12. We don't know what b of two
is, so let's keep going. We need to figure out b of two. If we use the same definition,
b of two is going to be equal to b of two minus one plus
12, so b of two minus one. That's b of one plus 12, but
we don't know what b of one is, so let's figure that out. B of one is equal to, well
here we can finally use this top clause, so b of one
is equal to negative seven. Now we can go and fill everything back in. If b of one is equal to negative
seven, then we know that this right over here is negative seven, and now we can figure out
that b of two is equal to negative seven plus 12
which is equal to five. If b of two is equal to
five, well then this is equal to five right over here,
and then now we know that b of three is equal to five plus 12 which is equal to 17. If we know that b of three is equal to 17, then we're ready to calculate
what b of four is going to be. B of four is now, it's b of three, which we figured out was 17 plus 12 which is equal to 29, and we are done.