Concrete mathematics : a foundation for computer science

This logic is seriously
floored.
3.30 x, = K2" + a-2", by induction; and X, is an integer.
A ANSWERS TO EXERCISES 495
3.31 Here’s an “elegant,” “impressive” proof that gives no clue about how
it was discovered:
lxj + lyj + lx + YJ = lx + 1YlJ + lx + YJ
6 1x-t ;12YlJ + lx+ il2Yl + $1
= [2x+ [ZyJ] = 12x1 + 12YJ.
But there’s also a simple, graphical proof based on the observation that we
need to consider only the case 0 6 x,y 0. Then l(x+ 1) =
l(x), and L(x) 6 l/2 for all x. When 0 6 x < 1, we have r(x) = x2/2 +x2/4 +
. . . =x2 andr(x+ 1)=(x- 1)2/2+(x+1)2/4+(x+1)2/8+~~~=~2+1.
Hencefix+l) =f(x)+l, whenO