Solve Inverse Problem: Find f(k) for sum_(k=1, to n) f(k) = F(n)

[Emilijo]

I have general function
sum_(k=1, to n) f(k) = F(n)

ex.) sum_(k=1, to n) k = F(n)
solution: F(n) = (1+n)n/2

But If I have inverse problem?:
ex.) sum_(k=1, to n) f(k) = n^2
how to get f(k) ?

Generaly, If I have F(n), how to get f(k)
for sum_(k=1, to n) f(k) = F(n)

Is there some formula or method to solve this?

 

[chiro]

I have general function
sum_(k=1, to n) f(k) = F(n)

ex.) sum_(k=1, to n) k = F(n)
solution: F(n) = (1+n)n/2

But If I have inverse problem?:
ex.) sum_(k=1, to n) f(k) = n^2
how to get f(k) ?

Generaly, If I have F(n), how to get f(k)
for sum_(k=1, to n) f(k) = F(n)

Is there some formula or method to solve this?

Hey Emilijo and welcome to the forums.

The first thing you should ask yourself is whether a unique inverse for a function is guaranteed to exist. If you do not place constraints on your function, do you think that you will have a unique function? What does this imply about the inverse?

 

[Emilijo]

For F(n)=n^2 in my example f(k)=2k-1,
because sum_(k=1,to n) (2k-1)= n^2 (wolfram alpha)
but what if I had a more complicated function, ex. sin(n)*n^2
Is there some formula to get f(k)?
In other words what function I have to sum when k goes from 1 to n
to get for example sin(n)*n^2, or generally to get function F(n)

 

[chiro]

For F(n)=n^2 in my example f(k)=2k-1,
because sum_(k=1,to n) (2k-1)= n^2 (wolfram alpha)
but what if I had a more complicated function, ex. sin(n)*n^2
Is there some formula to get f(k)?
In other words what function I have to sum when k goes from 1 to n
to get for example sin(n)*n^2, or generally to get function F(n)

The reason I asked you to consider whether the function is unique is because if its not, then you can never actually find a unique inverse for F(n). In general the answer is no which means that you won't be able to find a unique F(n) given the sum and the number of parameters in general.

As an example think of F(N) = ((N-1) MOD 3)-1 and G(N) = 1 - ((N-1) MOD 3). Now Calculate F(6) and G(6). F(6) = -1 + 0 + 1 + -1 + 0 + 1 = 0 and G(6) = 1 + 0 + -1 + 1 + 0 + -1 = 0. Both functions are completely different yet give the same answer.

Basically this has to do with the degrees of freedom you are given: the more degrees of freedom (in other words the higher the N) the more choices you can have and that translates eventually if you don't restrict things yourself to lots more functions.

 

[blue_raver22]

The inverse problem in this case is to find the function f(k) that satisfies the given equation sum_(k=1, to n) f(k) = F(n). This is a common problem in mathematics and there are various methods to solve it.

One approach is to use the method of finite differences. This involves taking the differences between consecutive terms in the given equation to obtain a new equation. By repeating this process, we can eventually obtain a linear equation in terms of f(k). Solving this equation will give us the function f(k).

Another approach is to use the method of generating functions. This involves converting the given equation into a generating function and then using techniques from generating function theory to obtain the function f(k).

There are also other methods such as the method of partial sums, the method of undetermined coefficients, and the method of interpolation that can be used to solve inverse problems of this form.

In summary, there are various methods that can be used to solve inverse problems in mathematics. The choice of method will depend on the specific equation and the tools available.