By R. Graham, D. Knuth, O. Patashnik
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25 ) If it’s geometric, there should be a geometric proof. 3 MANIPULATION OF SUMS 33 Ah yes, this formula was drilled into me in high school. ) The right-hand side can be remembered as the first term included in the sum minus the first term excluded (the term after the last), divided by 1 minus the term ratio. That was almost too easy. Let’s try the perturbation technique on a slightly more difficult sum, S, = x k2k O We must be careful, as always, not to divide by zero. The summationfactor method works whenever all the a’s and all the b’s are nonzero. 28 SUMS Let’s apply these ideas to a recurrence that arises in the study of “quicksort,” one of the most important methods for sorting data inside a computer. 12) for n > 0. k=O Hmmm. This looks much scarier than the recurrences we’ve seen before; it includes a sum over all previous values, and a division by n. Trying small cases gives us some data (Cl = 2, Cl = 5, CX = T) but doesn’t do anything to quell our fears. 1 + Qn-l)/Qn-2, Assume that Q,, # 0 for all n 3 0. Hint: QJ = (1 + oc)/(3. 9 Sometimes it’s possible to use induction backwards, proving things from n to n - 1 instead of vice versa! For example, consider the statement P(n) : x1 . x, 6 x1 +. . , x,30. n ( This is true when n = 2, since (x1 +xJ)~ -4~1x2 = (x1 -xz)~ 3 0. a b C 10 By setting x,, = (XI + ... + x,~l)/(n - l), prove that P(n) implies P(n - 1) whenever n > 1. Show that P(n) and P(2) imply P(2n). Explain why this implies the truth of P(n) for all n.