If x = (2 + 
) n , then the value of x – x 2 + x[x], where [·] denotes the greatest integer function, is equal to –
Text Solution
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x – x 2 + x[x] = x – x (x – [x]) = x – x {x} = x (1 – {x})
x = (2 + 
) n = n C 0 2 n + n C 1 2 n–1 
+ n C 2 2 n–2 ( 
) 2 + …
Let x 1 = (2 – 
) n = n C 0 2 n – n C 1 2 n–1 
+ n C 2 2 n–2 ( 
) 2 + …
x + x 1 = 2 ( n C 0 2 n + n C 2 2 n–2 · ( 
) 2 + …)
= Even integer.
Clearly x 1 ∈ (0, 1) ∀ n ∈ N.
⇒ [x] + {x} + x 1 = Even integer
⇒ {x} + x 1 = Integer
{x} ∈ (0, 1), x 1 ∈ (0, 1)
⇒ {x} + x 1 ∈ (0, 2)
⇒ {x} + x 1 = 1
⇒ x 1 = 1 – {x}
⇒ x(1 – {x}) = x · x 1 = (2 + 
) n (2 – 
) n = 1.
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