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Sequence-solving requires a great deal of mental activity -- it builds the knowledge base, pattern recognition, problem-solving, neuronal plasticity, memory and general intelligence. Here's a wonderful mini-challenge involving sequence-solving and a formula which can be used as a shortcut or a hack. Braintenance is a full-time occupation -- the more of it that you consciously, the more your uber-mind (i.e., superconscious) "takes the wheel" and replays the exercises. More than this, it tends to expand upon them, and experiment with them. Q: Find the next number in the following arithmetic series/sequence: 1, 2, 6, 24, 120, 720.... A: 5,040 Explanation : The numbers in the above sequence are factorials of the integers in order, i.e., 1!, 2!, 3!, 4!, 5!, 6!. The next one in the sequence would be 7!, which equals 5,040. Shortcut : The formula for estimating any n! is set forth below. If you memorize it, you can amaze your friends at parties and win bets at bars [although I ...

Image via Wikipedia In our last posting, we discussed the notion of asymptotes , and I posed two problems for your consideration. One involved the eventual (but unreachable) sum of a convergent series of numbers, and the other involving a ever-more troubling fraction. You can quickly refresh your memory by clicking on http://braintenance.blogspot.com/2011/09/asymptotes-closer-but-never.html , and by then hitting your browser's "BACK" button. The answers are unsatisfying, but they were promised: 1) In adding the sum of the series 1 + 1/2 + 1/4 + 1/8....and so forth, the sum will eventually approach, but never quite reach a limit of 2. 2) In dividing (n-1)/n, as n increases, the value of the expression approaches, but never reaches 1. There are examples of this type of complex conundrum in nature, in such things as trying to solve 22/7 (which is a never-ending decimal), and in determining the halflives of certain radioactive materials (isotopes), where one half of the ma...

Imagine trying to get to a finish line where with each leap you halve the distance remaining between you and your objective. Sounds good at first – but after a short period, you will realize that you can never actually get all the way there. You get closer and closer, halving the distance with each leap, but you won’t quite make it. Close, but no cigar. Your approach to the finish line is asymptotic . A mathematical limitation makes the intuitively simple task into the impossible conundrum. No matter how assiduously you proceed, you can merely cut the distance in half – even after a (theoretically) infinite number of leaps you cannot bridge that gap. The early leaps are the most productive…however, with each successive leap, your dilemma becomes clearer, and you become more frustrated . You have come up against a limiting mathematical constraint. The irony of this predicament is that although the objective is fixed, it might just as well be a moving target… retreating in smaller inc...