If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Algebra (all content)>Unit 4

Lesson 4: Constructing geometric sequences

# Explicit & recursive formulas for geometric sequences

Sal finds an explicit formula of a geometric sequence given the first few terms of the sequences. Then he explores equivalent forms the explicit formula and finds the corresponding recursive formula.

## Want to join the conversation?

• At What exponent property is Sal using to make (1/2)^(n-1) = [{(1/2)^n} * {(1/2)^-1}]
Has he made a video that could offer more examples of this? Or a link to a webpage.
Thank you very much
• How do I type in the answer for example in 2160 * (1/6) ^n-1 format? The n will power up but not the -1? How should I punch that in my phone?
• You need to put the n-1 in parentheses
2160 * (1/6) ^(n-1)
Otherwise the system sees (2160 * (1/6) ^n) - 1 due to operator precedence (order of operations).
• shouldn't the 1/2 be in parenthesis? should read (1/2)^(n-1)?
• If the number before the 1/2 is a variable, you would need to put parentheses over the 1/2, but otherwise you don't have to.
• What exactly is a recursive function?
• Some (or maybe all, I don't know for certain) functions have a recursive form, which states what kinds of outputs you will get for certain inputs. In this example, If n = 1, then our output, g(n), or g(1) in this case, is 168. For any whole number more than one, The output is 1/2 of the output of itself minus 1. g(2) = 1/2 * g(1), which we know is 168. Therefore, g(2) equals 84. g(3) equals half g(2), which is 1/2* g(1).Therefore, g(3)=1/2*(1/2*g(1)), or 42.
• I have an issue. In my homework, I have a sequence that, as I understand it, is neither arithmetic or geometric. (There has to be a constant, right)?

The sequence I'm facing is: 1,2,6,24,120,720... (this is raising x2,x3,x4,x5,x6,x7...)
I can see that there is some relationship - but I don't understand it. Even after hours on Khan Academy.

• This is a question,in general,How do you know when to use an Explicit or Recursive equation to solve a problem?
• Both equations require that you know the first term and the common ratio. Since you need the same information for both, ultimately it comes down to which formula best suits your needs.

The recursive formula requires that you know the term directly before the term you are looking to find. Therefore, if you are looking for a term that is within close proximity (ie find the 4th given the 1st) the recursive will probably be easiest and require less work.

The explicit formula will allow you to find any term without having to work through each consecutive term. This makes it particularly useful for finding terms that are a good way down the sequence (ie. find the 108th term).
• On the practice, how do you make "n-1" into one exponent because when I try to type it all into one exponent it wont work. :(
• On a side note: If you got a negative constant ratio, don't forget to wrap it as well.
Like this: a * (-8)^(n-1)
• so if the sequence was 3,6,12 would the equation be g(22) = 3 x 2^21. The final solution should be g(22)= 3 x 2097152 which is g(22) = 6291456?
• Merry christmas to everyone who reads this. I hope you have a happy holiday.
• Have a happy holiday as well!
• For one of the practice problems (Practice: Explicit formulas for geometric sequences) it says:
Haruka and Mustafa were asked to find the explicit formula for 4, 12, 36, 108
Haruka said g(n)= 4*3^n
Mustafa said g(n)= 4*4^n-1
the answer was that both of them were incorrect but I do not understand why that is the case. Could you explain how neither Mustafa or Haruka is correct?
• "n" represents the term
If n=1: g(1)=4*3^1 = 4*3 = 12
The first term is suppose to be 4, not 12.

Now for Mustafa:
If n=1: g(1) = 4*4^(1-1) = 4*4^0 = 4^1 = 4
so first term is ok.
If n=2: g(2) = 4*4^(2-1) = 4*4 = 16
The 2nd term is suppose to be 12, not 16.

The correct version would be: g(n)= 4*3^(n-1)

Hope this helps.