Let a n = ∫ - 1 n 1 + x 2 + x 2 3 + … + x n - 1 n d x for every n ∈ N . Then the sum of all the elements of the set n ∈ N : a n ∈ 2 , 30 is _________.

Question

Let a n = &#8747; - 1 n 1 + x 2 + x 2 3 + &#8230; + x n - 1 n d x for every n &#8712; N . Then the sum of all the elements of the set n &#8712; N : a n &#8712; 2 , 30 is _________.

Quantrex Academy · Accepted Answer

Correct answer: 0

Given a n = &#8747; - 1 n 1 + x 2 + x 2 3 + &#8230; . + x n - 1 n d x = x + x 2 2 2 + x 3 3 2 + &#8230; &#8230; + x n n 2 - 1 n i.e. a n = n + 1 1 2 + n 2 - 1 2 2 + n 3 + 1 3 2 + n 4 - 1 4 2 + &#8230; + n n + - 1 n + 1 n 2 Here a 1 = 2 , &#160; a 2 = 2 + 1 1 + 2 2 - 1 2 = 9 2 , a 3 = 4 + 2 + 28 9 = 100 9 , a 4 = 5 + 15 4 + 65 9 + 255 16 &#62; 31 . &#8757; &#160; a n &#8712; 2 , 30 , so the required set is 2 , 3 . &#8756; sum of all elements = 5 .

Let a n = ∫ - 1 n 1 + x 2 + x 2 3 + … + x n - 1 n d x for every n ∈ N . Then the sum of all the elements of the set n ∈ N : a n ∈ 2 , 30 is _________.

Correct answer

Step-by-step solution

More from Definite Integration

Let a n = &#8747; - 1 n 1 + x 2 + x 2 3 + &#8230; + x n - 1 n d x for every n &#8712; N . Then the sum of all the elements of the set n &#8712; N : a n &#8712; 2 , 30 is _________.

Correct answer

Step-by-step solution

More from Definite Integration

Let a n = ∫ - 1 n 1 + x 2 + x 2 3 + … + x n - 1 n d x for every n ∈ N . Then the sum of all the elements of the set n ∈ N : a n ∈ 2 , 30 is _________.