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## Algebra (all content)

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

Lesson 4: Constructing geometric sequences

# Converting recursive & explicit forms of geometric sequences

Sal solves the following problem: The explicit formula of a geometric sequence is g(x)=9*8^(x-1). Find the recursive formula of the sequence. Created by Sal Khan.

## Want to join the conversation?

• At , why does x have to be a positive integer? •  This is just the definition of the function. While you can still figure out the result of g(x) for non-positive integers, the function is not defined for these values. Imagine the function g(x) is used to determine how many grams of soap you would need to clean x plates. You could still determine the result of the function for fractional or negative or complex values of x, but these values wouldn't be within the domain of the problem, since you can't have "-3.4 plates".
• On the practise, KA will not allow me to input to the power of n - 1 Is there any work around because I am very bad at converting it?! • So then how do you convert a recursive formula (with the information of the first g(x) number) into an explicit formula? • Good question!
Well, the key pieces of information in both the explicit and recursive formulas are the first term of the sequence and the constant amount that you change the terms by, aka the common ratio (notice: the name "common ratio" is specific to geometric sequences, the name that applies to arithmetic seq. is "common difference") .
For example, you have the recursive formula:
`g(1)=9`
`g(x)=g(x-1)*(8)`
9 is the first term of the sequence, and 8 is the common ratio.
An explicit formula is structured as: `g(x)=(1st term of seq.)*(common ratio)^(x-1)`
Substitute for the variables, and you get the explicit formula: `g(x)= 9*8^(x-1)`

And there you have it!
Feel free to ask any more questions if you would like me to clarify my answer. ^-^
*Edit: I forgot to mention that the first term of a sequence is called the initial value.
• What are some practical applications that we could use recursive formula for instead of explicit formula? • Hey guys, I'm SUPER confused by these kind of problems:

{ f(1)=4
{ f(n)=f(n−1)⋅(−0.5)

Find an explicit formula for f(n).

f(n) = ?

In these problems, when you're given the answer/hints, the format is change to:
"f(n)=4⋅0.5^(n-1)" • Why would one use the recursive? The explicit is so much easier and straightforward. • Where can I find the formal syntax/grammar rules for defining these functions/sets/sequences? Sal gives several versions, which I love, but some seem more math-like than others. I'm new to this and I suspect that this shows up a lot in math. • when would you ever write or use a formula in its recursive form? I mean, by saying f(x) = f(x-1), you just add that -1 back by performing an operation on it, for e.g, *8, isn't it easier just to say that f(x) = f(x), which is already self evident?
(1 vote) • Yes, I believe what you are saying is correct. I believe Sal (and most every math teacher) is identifying the function. "Function" is used interchangeably with "f(x)." It's like saying: "The function is f(x-1)" or "f(x) is f(x-1)." Kinda like how we say "PIN number" which literally means "personal identification number number." but hearing the word "number" is actually helpful because it helps us recognize the term faster.
• I was inspired by these formulas to make an explicit and recursive formula for the remaining debt after payment per term on a mortgage, with a fixed payment per term plan. It seems I've managed to make a recursive one that works:

n = term
i = interest
p = terms per interest period (i.e. 12 terms per year)
R = remaining debt
a = original amount of the debt
m = monthly payment

Recursive formula R(n):
R(1) = a
R(n) = R(n-1)*(i/p+1)–m

I've been trying for over an hour now though, to convert this into an explicit one, with no luck. It seems that for a formula on the given issue to work, it is necessary to refer back to the previous term, and that you can't make explicit formulas for any sequence, even if you can make a recursive one. It would make sense, as I can't really see the value of recursive formulas if explicit ones always work. Explicit ones are a lot simpler and faster. Would I be right to make these conclusions?

Also interested to hear any feedback on the recursive formula. Does it check out? Could it be simpler?  