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Giving a general term of a recursive function, and upper bound for it

Let a constant $B \ge 1$, and let $l_1 = 0$, $b_1 = 0$ be the values of $l$ and $b$ (respectively) at time $t = 1$. Let $l_{t+1} = l_t + 1$ if $b_i < B$, and $l_{t+1} = l_t$ otherwise Let ...
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On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that $$ A_0 = A_1 = 0;\quad A_n = ...