Some Guidance

[JeffM]

I was asked a question by private message. I have figured out this much of an answer.

Answering the question involves solving a system of n + 1 linear equations subject to constraints.

[MATH]c > 0, \ 1 \le j \le n + 1 \implies 0 \le a_j, 1 < p_j, \text {and } \\ \displaystyle a_jp_j = c + \sum_{j=1}^{n+1} a_j.[/MATH]The remaining questions are

Is at least one solution guaranteed? If so, it it unique? Finally, is it practicable to implement the solution in excel?

I suspect this is a snap using matrix algebra, but I don't know matrix algebra.

If it helps I have worked out the n = 1 case. A unique solution fitting the constraints exists. It can easily be computed with a hand calculator.

[MATH]a_2 = \dfrac{cp_1}{p_1p_2 - (p_1 + p_2)} \text { and }\\ a_1 = \dfrac{cp_2}{p_1p_2 - (p_1 + p_2)}.[/MATH]And it is easy to show that [MATH]p_1p_2 - (p_1 + p_2) > 0.[/MATH]
I'll work on an induction step tomorrow, but if someone can tell me a quick solution using linear algebra, that would perhaps save me a lot of unnecessary work.

 

[Steven G]

Jeff, can you please try to explain the problem as I am surely missing something?

 

[JeffM]

I suspect you are missing nothing and that this is is an easy problem with the right tools. I simply may not own the right tools.

If n = 1, the system looks like

[MATH]a_1p_1 = c + a_1 + a_2 \text { and } a_2p_2 = c + a_1 + a_2, \text { where}\\ c > 0,\ a_1 \ge 0 \le a_2, \text { and } p_1 > 1 < p_2.[/MATH]To satisfy the constraints, the solution requires that [MATH]p_1p_2 - (p_1 + p_2) > 0.[/MATH]
That can be proved readily.

The unique solution can then be found by first year algebra. So that problem is not at all hard.

My difficulties are that I need to explain how to implement a solution in excel for large n, which won't be hard if I can find a general solution. The case where n = 1 looks promising for a nice clean general solution. My other difficulty so far is I am not sure yet how to extend a result true for a system of k + 1 equations in k + 1 variables to a system of k + 2 equations in k + 2 variables. I assume it can be done, and I have at least one idea on how to do it. But I am not doing this for my own amusement. I am doing it in response to an OP who for some reason sent his problem to me as a private message. I have spent quite a bit of time trying to decipher what his problem was really about. It turns out that it reduces to what I showed in my original post. If the original problem had been in the forum, I suspect someone would have already said if there is an easy way to go.

Does this help?

 

[JeffM]

Actually, the p's must meet more restrictive conditions.

 

[JeffM]

According to the conditions of the problem, it is given that p₁ > 1 < p₁, but that is not sufficient to guarantee that a solution exists nor that both a₁ and a₂ are positive. To meet those constraints, it is also necessary that

[MATH]p_1p_2 > p_1 + p_2.[/MATH]
Having slept on it, I am going to try the induction step.