Let x and x denote the fractional part of x and the greatest integer ≤ x respectively of a real number x . if ∫ 0 n x d x , ∫ 0 n x d x and 10 n 2 - n , ( n ∈ N , n 1 ) are three consecutive terms of a G.P. then n is equal to__

Question

Let x and x denote the fractional part of x and the greatest integer ≤ x respectively of a real number x . if ∫ 0 n x d x , ∫ 0 n x d x and 10 n 2 - n , ( n ∈ N , n > 1 ) are three consecutive terms of a G.P. then n is equal to__,

MARKS App · Accepted Answer

∫ 0 n x d x = n ∫ 0 1 x d x = n x 2 2 0 1 = n 2 and ∫ 0 n x d x = ∫ 0 n x - x d x = x 2 2 0 n - ∫ 0 n x d x = n 2 2 - n 2 Now, n 2 , n 2 - n 2 and 10 n 2 - n are in Geometric progression So, n 2 - n 2 2 = n 2 · 10 n 2 - n ⇒ n 2 ( n - 1 ) 2 4 = 5 . n 2 ( n - 1 ) ⇒ n - 1 = 20 ⇒ n = 21 ∵ n ≠ 1

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