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# Worked example: using recursive formula for arithmetic sequence

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

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• when you put b(1) you get=12, in the data b(1)=-7, I didn't get it?
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.
• Is there a faster way to find terms using the recursive formula?
• you can convert a recursive to explicit very easily
• 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!
• 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.
• 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?
• Is there a quicker way to solve this type of problem?
• you can convert the recursive formula to an explicit formula
• 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 ?
• 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.
(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.
• 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.
• Would you do the same if the sequence was decreasing?
(1 vote)
• Yes you just end up with a negative sign
• 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.