Let f ⁡ n = 1 3 + 3 n 100 n , where [ n ] denotes the greatest integer less than or equal to n . Then ∑ n = 1 56 f n is equal to

Step-by-step solution

Given f n = 1 3 + 3 n 100 n = 9 n + 100 300 n If 9 n + 100 300 < 1 ⇒ 9 n + 100 < 300 ⇒ n < 200 9 ⇒ n = 22 We know that for a number 0 ⩽ x < 1 ,   x = 0 ⇒ f ( n ) = 0 ,   ∀   1 ⩽ n ⩽ 22 Again, if 9 n + 100 300 < 2 ⇒ 9 n + 100 < 600 ⇒ n = 500 9 ⇒ n = 55 We know that for a number 1 ⩽ x < 2 ,   x = 1 ⇒ f ( n ) = 1 ,   23 ⩽ n ⩽ 55 And, f ( n ) = 2 if n = 56 Thus, ∑ n