Explicație pas cu pas:
legea de compoziție asociativă:
[tex]x \circ y = \frac{x y}{x+y} \\ [/tex]
a)
[tex]2 \circ 2=\frac{2 \times 2 }{2 + 2} = \frac{4}{4} = 1\\ [/tex]
b)
legea ○ este asociativă:
[tex]x \circ y \circ z = (x \circ y) \circ z = \frac{(x \circ y) \times z}{(x \circ y) + z} \\[/tex]
[tex]= \frac{\frac{x y}{x+y} \times z}{\frac{x y}{x+y} + z} = \frac{\frac{x yz}{x+y}}{\frac{xy + z(x + y)}{x+y}} \\ [/tex]
[tex]= \frac{xyz}{xy + yz + xz} = {\left( \frac{xy + yz + xz}{xyz} \right) }^{ - 1} \\ [/tex]
[tex]= {\left( \frac{xy}{xyz} + \frac{yz}{xyz} + \frac{xz}{xyz} \right) }^{ - 1} = {\left( \frac{1}{z} + \frac{1}{x} + \frac{1}{y} \right) }^{ - 1} \\ [/tex]
[tex]= {\left( {x}^{ - 1} + {y}^{ - 1} + {z}^{ - 1}\right) }^{ - 1} \\ [/tex]
c) din b)
→
[tex]\frac{1}{2} \circ \frac{1}{3} \circ \frac{1}{4} \circ \ldots \circ \frac{1}{10} = \\ [/tex]
[tex]= {\left( {\left( \frac{1}{2} \right)}^{ - 1} + {\left( \frac{1}{3} \right)}^{ - 1} + {\left( \frac{1}{4} \right)}^{ - 1} +...+ {\left( \frac{1}{10} \right)}^{ - 1}\right) }^{ - 1} \\ [/tex]
[tex]= {\left(2 + 3 + 4 + ... + 10\right) }^{ - 1} = {\left( \frac{10 \times 11}{2} - 1 \right) }^{ - 1} \\ [/tex]
[tex]= {54}^{ - 1} = \frac{1}{54} \\ [/tex]