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. the

Question

Let x and x denote the fractional part of x and the greatest integer &#8804; x respectively of a real number x . if &#8747; 0 n x d x , &#8747; 0 n x d x and 10 n 2 - n , ( n &#8712; N , n &#62; 1 ) are three consecutive terms of a G.P. then n is equal to__

Quantrex Academy · Accepted Answer

Correct answer: 0

&#8747; 0 n x d x = n &#8747; 0 1 x d x = n x 2 2 0 1 = n 2 and &#8747; 0 n x d x = &#8747; 0 n x - x d x = x 2 2 0 n - &#8747; 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 &#183; 10 n 2 - n &#8658; n 2 ( n - 1 ) 2 4 = 5 . n 2 ( n - 1 ) &#8658; n - 1 = 20 &#8658; n = 21 &#8757; n &#8800; 1

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__

Correct answer

Step-by-step solution

More from Definite Integration

Let x and x denote the fractional part of x and the greatest integer &#8804; x respectively of a real number x . if &#8747; 0 n x d x , &#8747; 0 n x d x and 10 n 2 - n , ( n &#8712; N , n &#62; 1 ) are three consecutive terms of a G.P. then n is equal to__

Correct answer

Step-by-step solution

More from Definite Integration

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__