**Question:**

Proove [X]=[X/2]+[(X+1)/2] Where, [ ] is greatest integer function

2

1

So for 5.3 the greatest integer is 5 and for 5.7 also greatest integer is 5.

[X/2] = [X]/2 and

[(X+1)/2] = [X]/2 + 1 if [X} is odd

and = [X]/2 when [X] is even.

If [X] is odd then

[X] = [X/2] + [X/2] + 1 = [X/2]+[(X+1)/2]

if [X] is even

[X] = [X/2] + [X/2] = [X/2] + [(X+1)/2] (as it is equal to [X/2] if [X] is even)

Thus for both cases

[X]=[X/2]+[(X+1)/2]

Hence Proved.