Math Background. Floors & Ceilings. For any real number x , we denote the greatest integer less than or equal to x by x read “ the floor of x ” For any real number x , we denote the least integer greater than or equal to x by x read “ the ceiling of x ”
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We need it for computing the running time of a given algorithm.
1 + 2 + … + n + n+1 = (1 + 2 + …+ n) + (n+1)
= n(n+1)/2 + n+1 = [n(n+1) + 2(n+1)]/2
= (n+1)(n+2)/2 = (n+1)(n+1 + 1) / 2
a0 = 1 = (a1 - 1)/(a - 1) = 1
a0 + a1 + … + an+1 = a0 + a1 + … + an + an+1
= (an+1 - 1)/(a - 1) + an+1 = (an+1+1 - 1)/(a - 1)