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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!
(1 vote)
• 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?
• 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?
• 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)
• 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.
• Is there a quicker way to solve this type of problem?
• you can convert the recursive formula to an explicit formula
• 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.
• convert to explicit form: a(i) = -7 + 12(n-1)
a(100) = -7 + 12(99)
a(100) = -7 + 1188
a(100) = 1181
Hope this helps!
• 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 ?