If $p,q$, and $r$ are terms of an arithmetic progression are also in a geometric progression, then find the common ratio of the geometric progression in terms of $p,q$, and $r$.
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Hint: Let $p,q,r$ be the terms. Let $x$ be the ratio in the geometric progression, $y$ be the increment in the arithmetic progression. $x=\frac qp, y=q-p$ Then the terms are $p, px, px^2$ and also $p, p+y, p+2y$. Can find two equations for $x$ and $y$? Added: You have $p+y=px, p+2y=px^2$. We have two equations in two unknowns if we regard $p$ as a parameter. Subtracting, $y=px^2-px$. Dividing, $x=\frac rq=\frac {p+2y}{p+y}$. I suspect the desired answer is the solution for $x$ of these two. |
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