For a real number x,[x] denotes the greatest integer less than or equal to x . Then the value of [ 1 2 ]+ [ 1 2 + 1 100 ]+ [ 1 2 + 2 100 ]+ [ 1 2 + 3 100 ]+ .+ [ 1 2 + 99 100 ]=

Step-by-step solution

The sum is S = [ 1 2 ]+ [ 1 2 + 1 100 ]+ + [ 1 2 + 99 100 ] . Rewriting 1 2 as 50 100 , each term becomes [ 50+k 100 ] for k = 0, 1, 2, , 99 . When 50+k 100 For k = 0, 1, , 49 , the term equals 0 . When 1 50+k 100 For k = 50, 51, , 99 , the term equals 1 . There are 50 terms in each group. Summing gives S = 50 0 + 50 1 = 50 . 50