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.
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.
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