Question

If x = (2 +$\sqrt{3}$)ⁿ, then the value of x – x² + x[x], where [·] denotes the greatest integer function, is equal to –

• A. 1
• B. 2
• C. 2²ⁿ
• D. 2ⁿ

Answer 1

Answer: (A)

1

Answer 2

x – x² + x[x] = x – x (x – [x]) = x – x {x} = x (1 – {x})

x = (2 +$\sqrt{3}$)ⁿ = ⁿC₀ 2ⁿ + ⁿC₁ 2^n–1$\sqrt{3}$+ ⁿC₂ 2^n–2 ($\sqrt{3}$)² + …

Let x₁ = (2 –$\sqrt{3}$)ⁿ = ⁿC₀ 2ⁿ – ⁿC₁ 2^n–1$\sqrt{3}$+ ⁿC₂ 2^n–2 ($\sqrt{3}$)² + …

x + x₁ = 2 (ⁿC₀ 2ⁿ + ⁿC₂ 2^n–2 · ($\sqrt{3}$)² + …)

= Even integer.

Clearly x₁ Ī (0, 1) " n Ī N.

Ž [x] + {x} + x₁ = Even integer

Ž {x} + x₁ = Integer

{x} Ī (0, 1), x₁ Ī (0, 1)

Ž {x} + x₁ Ī (0, 2)

Ž {x} + x₁ = 1

Ž x₁ = 1 – {x}

Ž x(1 – {x}) = x · x₁ = (2 + $\sqrt{3}$)ⁿ (2 –$\sqrt{3}$)ⁿ = 1.