Question
If one G.M. $G$ and two arithmetic means $p$ and $q$ be inserted between any two given numbers then show that $G^{2}=(2 p-q)(2 q-p)$.
Step 1
Given that $G$ is the geometric mean between $a$ and $b$, we have \[G = \sqrt{ab}\] Squaring both sides, we get \[G^{2} = ab \tag{1}\] Show more…
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If $p, m,$ and q form an arithmetic sequence, it can be shown that $m=(p+q) / 2 .$ (See Exercise 63.) The number $m$ is the arithmetic mean, or average, of $p$ and q. Given two numbers p and q, if we find k other numbers $m_{1}, m_{2}, \ldots, m_{k}$ such that $$ p, m_{1}, m_{2}, \dots, m_{k}, q $$ forms an arithmetic sequence, we say that we have "inserted $k$ arithmetic means between $p$ and $q$." Insert three arithmetic means between 4 and 12 .
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If $p, m,$ and q form an arithmetic sequence, it can be shown that $m=(p+q) / 2 .$ (See Exercise 63.) The number $m$ is the arithmetic mean, or average, of $p$ and q. Given two numbers p and q, if we find k other numbers $m_{1}, m_{2}, \ldots, m_{k}$ such that $$ p, m_{1}, m_{2}, \dots, m_{k}, q $$ forms an arithmetic sequence, we say that we have "inserted $k$ arithmetic means between $p$ and $q$." Insert four arithmetic means between 4 and 13 .
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