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## Algebra 1

### Course: Algebra 1>Unit 9

Lesson 1: Introduction to arithmetic sequences

# Worked example: using recursive formula for arithmetic sequence

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?
(12 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.
(13 votes)
• Is there a faster way to find terms using the recursive formula?
(5 votes)
• you can convert a recursive to explicit very easily
(8 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.
(6 votes)
• Would the concept be the same if the variable contains roots? For example, instead of the original problem: b(n)=b(n-1)+12 can you solve it the same way if the problem looked more like b(√6)=b(√6 -1)+12?
(3 votes)
• No, you can't

Sequence n numbers has to be a positive INTEGER

Meaning the input can't contain a decimal(square root of 6 results in a decimal answer) or negative
(1 vote)
• 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)
• lebron
(3 votes)
• yes siii
(1 vote)
• 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)
• Just use A instead of random letters :/
(3 votes)
• Recursive formulas seem like a waste of time. Why not just always use explicit formulas instead?
(1 vote)
• There are some sequences such as the Fibonacci sequence that is easier to do with recursive formulas vs explicit formulas.
(4 votes)
• At , 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)

## 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.