Solving a series of system equations

Question

Let $P = <p_1, \ldots,p_k>$,
$Q = <q_1, \ldots,q_k>$ and PQ = v Solve for P and Q, where

1. $p_1q_1 + p_2q_2 + \ldots + p_kq_k = v$

2. $q_1 + q_2 + \ldots + q_k = t$

3. $P$ is an AP series with $a_1, a_1+d, \ldots, a_1+(k-1)d$

Answer 1

You have $k+2$ unknowns ($a$ and $d$ to determine $P$ and $q_1...q_k$ for $Q$) and only two equations. So you have $k$ degrees of freedom.

So you could for example make the following $k$ choices: $q_3=...=q_k=0$, and some values for $a$ and $d$ (non-zero). This leaves you with the following system of two equations with two unknowns:

$$q_1+q_2=t\\aq_1+(a+d)q_2 = v$$

Solve these for $q_1$ and $q_2$ and you have one of the infinitely many possible answers.