Let α denote the greatest integer ≤ α . Then 1 + 2 + 3 + . . . . . . . . . . . . . + 120 is equal to

Step-by-step solution

Given, 1 + 2 + 3 + . . . . . . . . . . . . . + 120 Now we now that, x = 1 ,   when   x ∈ [ 1 , 4 ) , x = 2 ,   when   x ∈ [ 4 , 9 ) and similarly x = 10 ,   when   x ∈ [ 100 , 120 ) Now using the above formula we get, E = 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + … ⇒ E = 3 × 1 + 5 × 2 + 7 × 3 + … . . + 19 × 9 + 10 × 21 ⇒ E = ∑ r = 1 10 2 r + 1 r ⇒ E = 2 ∑ r = 1 10 r 2 + r &#86