Question

ayudenme resolver el problema numero 3

Answer 1

a² -3a + 1 = 0

a₁ = 3/2 + √5/2

a₂ = 3/2 - √5/2

M = (a² + a⁻²)/(a³ - a⁻³)

M₁ = (a₁² + a₁⁻²)/(a₁³ - a₁⁻³)

M₁ = (a₁² + 1/a₁²)/(a₁³ - 1/a₁³)

a₁² = (3/2 + √5/2)² = 9/4 + 3√5/2 + 5/4 = 7/2 + 3√5/2

a₁³ = (3/2 + √5/2)³ = (3/2 + √5/2)(7/2 + 3√5/2) = 21/4 + 9√5/4 + 7√5/4 + 15/4 = 9 + 4√5

M₁ = (7/2 + 3√5/2 + 1/(7/2 + 3√5/2)/(9 + 4√5 - 1/(9 + 4√5))

M₁ = (7/2 + 3√5/2 + (7/2 - 3√5/2)/(49/4 - 45/4)/(9 + 4√5 - (9 - 4√5)/(81 + 80))

M₁ = (7/2 + 3√5/2 + 7/2 - 3√5/2)/(9 + 4√5 - 9 + 4√5)

M₁ = 7/(8√5) = (7/40)√5

M₂ = (a₂² + a₂⁻²)/(a₂³ - a₂⁻³)

M₂ = (a₂² + 1/a₂²)/(a₂³ - 1/a₂³)

a₂² = (3/2 - √5/2)² = 7/2 - 3√5/2

a₂³ = (3/2 - √5/2)³ = 9 - 4√5

M₂ = (7/2 - 3√5/2 + 1/(7/2 - 3√5/2)/(9 - 4√5 - 1/(9 - 4√5))

M₂ = (7/2 - 3√5/2 + (7/2 + 3√5/2)/(49/4 - 45/4)/(9 + 4√5 - (9 - 4√5)/(81 + 80))

M₂ = (7/2 - 3√5/2 + 7/2 + 3√5/2)/(9 - 4√5 - 9 - 4√5)

M₂ = 7/(-8√5) = -(7/40)√5

M₁ = 7/(8√5)

M₂ = -7/(8√5)

M₁ = -M₂

|M| = 7/(8√5)

M = ±7/(8√5)