Determaning a function from a progression?

• christian0710
In summary, the progression described in the conversation involves adding a new term (x-n) to the previous terms every time n increases by one. The goal is to turn this into a mathematical function of n. After some discussion, the function is determined to be x*∏k(x/k-1), with the Pochhammer symbol and Gamma function being related to its closed form.
christian0710
Hi,
If i have a progression that looks like thisn | progression | Mathmatical formula
--------------------------------------
1 | x(x-1) .....| x(x-n)
2 | x(x-2)(x-1) .....| x(x-n)(x-n-1)
3 | x(x-3)(x-2)(x-1)...| x(x-n)(x-n-1)(x-n-2))

Every time n increases by one a new term (x-n) is added to the previous terms. How do I turn this into a function of n? I know that as n increases then we get (x-(n-1))*(x-(n-2)) ... (x-(n-(n-1)))) n times but how do you express this as a mathematical function? A function that adds keeps adding n terms (x-n)(x-n-1)(x-n-2)etc... all the way down to the last term (x-n-n-1)

Can anyone help me or give me a hint about how to construct a function of n, such that the above progression holds true for any n we put into the function?

I know that x! = x*x-1*x-2*--*x-x+1

but saying f(n)= (x-n)! does not make sense since we have two variales (x and n) , I'm stuck and would really appreciate a hint :)

Last edited:
Wait I think i might have igured it out - or am i doing the math wrong?

I know that
k from 1 to n is defined as k = 1*2*3*4...n

Therefore it must be true that

k(x/k - 1) going from 1 to n is defined as

∏k(x/k-1) =( (1*x)/1 -1) * ((2*x)/2-2) * ((3*x)/3-3) = (x -1)(x-2)(x-3) etc.. p to n :) Is this corret usage of the mathematical multiplication operator ∏?

So the function for this progression is
x*∏k(x/k-1)

See the Pochammer symbol for this function and the related closed form thanks to the Gamma function :
x(x-1)(x-2)...(x-n)=Gamma (x+1) / Gamma(x+1-n)

1. How do you determine the function from a progression?

To determine the function from a progression, you need to first observe the pattern of the progression. Look for any consistent change in the values as the progression continues. This will give you a clue as to what type of function the progression may follow. Then, plug in the values of the progression into the potential function and see if they match. If they do, then you have successfully determined the function from the progression.

2. What is the difference between arithmetic and geometric progressions?

Arithmetic progressions have a constant difference between each term, while geometric progressions have a constant ratio between each term. In other words, in arithmetic progressions, the difference between two consecutive terms is the same, whereas in geometric progressions, the ratio between two consecutive terms is the same.

3. Can a progression have more than one possible function?

Yes, it is possible for a progression to have more than one possible function. This can happen when the values of the progression do not follow a clear pattern or when there are multiple patterns that can be observed in the progression. In such cases, it is important to consider the context of the progression and choose the most appropriate function.

4. What are some common types of functions that can be determined from a progression?

Some common types of functions that can be determined from a progression include linear, quadratic, exponential, and logarithmic functions. Other less common types include trigonometric, rational, and polynomial functions.

5. Are there any shortcuts or tricks to determine the function from a progression?

While there are no shortcuts or tricks to determine the function from a progression, there are some strategies that can make the process easier. These include looking for patterns, using algebraic techniques to manipulate the progression, and using graphing software to visualize the data. It is also helpful to have a good understanding of different types of functions and their characteristics.

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