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(A) Let the two numbers be $a$ and $b$.Since $p$ and $q$ are two arithmetic means between $a$ and $b$,the sequence $a, p, q, b$ is in Arithmetic Progr

Vedclass · Accepted Answer

$(2p - q)(2q - p)$

Vedclass · Answer

(A) Let the two numbers be $a$ and $b$.Since $p$ and $q$ are two arithmetic means between $a$ and $b$,the sequence $a, p, q, b$ is in Arithmetic Progression ($A$.$P$.).Let the common difference be $d$.Then $p = a + d$,$q = a + 2d$,and $b = a + 3d$.From these,$d = q - p$,so $a = p - d = p - (q - p) = 2p - q$.Also,$b = a + 3d = (2p - q) + 3(q - p) = 2p - q + 3q - 3p = 2q - p$.Since $G$ is the geometric mean between $a$ and $b$,$G^2 = ab$.Substituting the values of $a$ and $b$,we get $G^2 = (2p - q)(2q - p)$.

If $p$ and $q$ are two arithmetic means between two numbers and $G$ is the geometric mean between them,then $G^2 = \dots \dots$.

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