Predicting the Next Number
Date: 8/30/96 at 10:44:46 From: Anonymous Subject: Predicting the next number in a sequence When given a series of numbers and asked to predict the next number, what is the formula for doing so? Example: 2,5,12,23, ? This question appears on psychological exams, federal employment exams and many others. Is there a mathematical way of determining the next number in this series?
Date: 8/30/96 at 12:46:8 From: Doctor Jerry Subject: Re: Predicting the next number in a sequence First, if the first several terms of a sequence are given, there is no method for determining the general term. Suppose I'm given the numbers a, b, c, d and asked to determine the fifth and sixth terms of the sequence. I'll show that any number of different solutions is possible. I start by determining a polynomial p(x) = Ax^3 + Bx^2 + Cx + D such that p(1) = a, p(2) = b, p(3) = c, and p(4) = d. Then consider f(n) = p(n) + (n-1)(n-2)(n-3)(n-4) or g(n) = p(n) + (n-1)(n-2)(n-3)(n-4)(n-5). Notice that both f and g determine sequences whose first four terms are a, b, c, and d. Remaining terms are wildly different. This idea could be elaborated. You can, however, try to guess what was most likely in the mind of the person who made up the question. For the sequences 2,4,6,8,... 1,4,9,16,... I suppose most persons would say 10,12 or 25,36. For the sequence you gave, 2, 5, 12, 23, I noticed that 5 - 2 = 3, 12 - 5 = 7, and 23 - 12 = 11. Since 3, 7, and 11 can be viewed as the odd numbers, leaving every other one out, one could argue that the next two terms are 23 + 15 and 38 + 19. Other, more or less natural answers are possible. However, one has no choice but to accept whatever the text makers decree is the correct answer! -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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