Question

If one GM '' and two AM's r and s are inserted between two positive numbers, prove that G^2= (2r - 5)(2s - r).​

Answer 1

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$let \: the \: two \: numbers \: be \: a\: and \: b \: \\ then \\ G \sqrt{ab} = {G}^{2} = ab \\ alo \: p \: and \: q \: are \: two \: a.m.s \: between \: a \: and \: b \: \\ a \: p \: q \: b \: are \: in \: a.p. \\ p \times a = q - p \: and \: q - p = b - q \\ a \: = 2p - q \: andb = 2q - p \\ {G}^{2} = ab = (2p - q)(2p - q)$

$\huge\color{orange}\boxed{\colorbox{gray}{➼hope it works!!!}}$