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Question Number 98022 by Ar Brandon last updated on 11/Jun/20

D erive the relation between an A rithmetic M ean

and a G eometric M ean

\sqrt[n]{x_{1} x_{2} \dots x_{n}} ⩽ \frac{x_{1} + x_{2} + \cdot \cdot \cdot + x_{n}}{n} \forall n \in N^{*}, \forall (x_{1}, x_{2}, \dots x_{n}) \in {(R_{+}^{*})}^{n}

Answered by Rio Michael last updated on 11/Jun/20

The geometric mean G = \sqrt{a b}

The arithmetic mean A = \frac{a + b}{2}

A - G > 0

A - G = \frac{a + b}{2} + \sqrt{a b}

= \frac{a + b - 2 \sqrt{a b}}{2}

= \frac{{(\sqrt{a} - \sqrt{b})}^{2}}{2}

for a quadratic x^{2} - 2 A x + G^{2} = 0 where G^{2} = \sqrt{a b} and A = \frac{a + b}{2}

has roots x = A \pm \sqrt{A^{2} - G^{2}}

Commented by Ar Brandon last updated on 11/Jun/20

T hanks for your time . B ut actually, the question requires

us to derive the above relation, instead of solving for x .