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Enunciado

Solución

x + 1/x = √ 2 , 1 qué vale x ⁷⁷⁷ + ―― x ⁷⁷⁷

Este problema se puede resolver empleando las propiedades de las funciones simétricas.

Una función es simétrica, si intercambiando cualesquiera de sus variables el valor de la función no varía

x = α , 1/x = β

α + β = √ 2
α ∙ β = 1

Como es bien conocido, relaciones de Cardano-Vieta , α y β son raíces de la ecuación,

ξ ² − √ 2 ∙ ξ + 1 = 0

α = √ 2 /2 ∙ ( 1 + i ) = е^{i ∙ π/4}
β = √ 2 /2 ∙ ( 1 − i ) = е^{− i ∙ π/4}

Sólo resta evaluar α y β elevadas a la 777 potencia,

777 = 4 ∙ 194 + 1

е^{π ∙ 777/4} = е^{π ∙ 194} ∙ е^π/4 = е^{π ∙ 2} ∙ е^π/4 = е^π/4

∴ x ⁷⁷⁷= x

De dónde,

1 x ⁷⁷⁷ + ―― = √ 2 x ⁷⁷⁷

∎

English

Español

Statement

Answer

x + 1/x = √ 2 , 1 calculate x ⁷⁷⁷ + ―― x ⁷⁷⁷

This is a problem on symmetric functions.

A function is symmetric if the exchange of its variables does not alters its value

x = α , 1/x = β

α + β = √ 2
α ∙ β = 1

As it is well known, Vieta's formula , α and β are roots of the equation,

ξ ² − √ 2 ∙ ξ + 1 = 0

α = √ 2 /2 ∙ ( 1 + i ) = е^{i ∙ π/4}
β = √ 2 /2 ∙ ( 1 − i ) = е^{− i ∙ π/4}

Evaluating α and β to the 777 power,

777 = 4 ∙ 194 + 1

е^{π ∙ 777/4} = е^{π ∙ 194} ∙ е^π/4 = е^{π ∙ 2} ∙ е^π/4 = е^π/4

∴ x ⁷⁷⁷= x

Whence,

1 x ⁷⁷⁷ + ―― = √ 2 x ⁷⁷⁷

∎

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Tácticas matemáticas. Mathematical ploys.⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Enunciado

Solución

x + 1/x = √ 2 , 1qué vale x 777 + ―― x 777

x = α , 1/x = β α + β = √ 2 α ∙ β = 1

ξ 2 − √ 2 ∙ ξ + 1 = 0 α = √ 2 /2 ∙ ( 1 + i ) = е i ∙ π/4 β = √ 2 /2 ∙ ( 1 − i ) = е − i ∙ π/4

777 = 4 ∙ 194 + 1 е π ∙ 777/4 = е π ∙ 194 ∙ е π/4 = е π ∙ 2 ∙ е π/4 = е π/4∴ x 777 = x

1 x 777 + ―― = √ 2 x 777

Statement

Answer

x + 1/x = √ 2 , 1calculate x 777 + ―― x 777

x = α , 1/x = β α + β = √ 2 α ∙ β = 1

ξ 2 − √ 2 ∙ ξ + 1 = 0 α = √ 2 /2 ∙ ( 1 + i ) = е i ∙ π/4 β = √ 2 /2 ∙ ( 1 − i ) = е − i ∙ π/4

777 = 4 ∙ 194 + 1 е π ∙ 777/4 = е π ∙ 194 ∙ е π/4 = е π ∙ 2 ∙ е π/4 = е π/4∴ x 777 = x

1 x 777 + ―― = √ 2 x 777

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x + 1/x = √ 2 ,

1
qué vale x ⁷⁷⁷ + ――
x ⁷⁷⁷

x = α , 1/x = β

α + β = √ 2
α ∙ β = 1

ξ ² − √ 2 ∙ ξ + 1 = 0

α = √ 2 /2 ∙ ( 1 + i ) = е^{i ∙ π/4}
β = √ 2 /2 ∙ ( 1 − i ) = е^{− i ∙ π/4}

777 = 4 ∙ 194 + 1

е^{π ∙ 777/4} = е^{π ∙ 194} ∙ е^π/4 = е^{π ∙ 2} ∙ е^π/4 = е^π/4

∴ x ⁷⁷⁷= x

1
x ⁷⁷⁷ + ―― = √ 2
x ⁷⁷⁷

x + 1/x = √ 2 ,

1
calculate x ⁷⁷⁷ + ――
x ⁷⁷⁷

x = α , 1/x = β

α + β = √ 2
α ∙ β = 1

ξ ² − √ 2 ∙ ξ + 1 = 0

α = √ 2 /2 ∙ ( 1 + i ) = е^{i ∙ π/4}
β = √ 2 /2 ∙ ( 1 − i ) = е^{− i ∙ π/4}

777 = 4 ∙ 194 + 1

е^{π ∙ 777/4} = е^{π ∙ 194} ∙ е^π/4 = е^{π ∙ 2} ∙ е^π/4 = е^π/4

∴ x ⁷⁷⁷= x

1
x ⁷⁷⁷ + ―― = √ 2
x ⁷⁷⁷