Question

Prove that:
$(\sqrt{5}+1)^{5}-(\sqrt{5}-1)^{5}=352$

Answer 1

Answer:

352

Step-by-step explanation:

Hi,

Expanding (√5 + 1)⁵ using binomial expansion we get

(√5 + 1)⁵ = ⁵C₀(√5)⁵ + ⁵C₁(√5)⁴ + ⁵C₂(√5)³ + ⁵C₃(√5)² + ⁵C₄(√5)

             + ⁵C₅-------------(1)

(√5 - 1)⁵ = ⁵C₀(√5)⁵ - ⁵C₁(√5)⁴ + ⁵C₂(√5)³ - ⁵C₃(√5)² + ⁵C₄(√5)

             - ⁵C₅-------------(2)

Subtracting (1) - (2), we get

(√5 + 1)⁵ - (√5 - 1)⁵

= 2[⁵C₁(√5)⁴  + ⁵C₃(√5)² + ⁵C₅]

= 2[5*25 + 10*5 + 1]

= 2[125 + 50 + 1]

= 2[176]

= 352

Hope, it helps !