 0

# What numbers fill the spaces in this sequence 45 39 30 .space..space. 24 and if possible what is the formula?

This sequence is puzzling, particularly because the steps initially increase in magnitude; and yet the sequence then takes three steps just to fall from 30 to 24, a surprise which runs counter to our intuitions. Alternative reasons for this odd behaviour could be: * The sequence isn't a straightforward function, but rather involves some recursive relationship between terms. * The series is an elementary function, but it isn't using the counting numbers as its input. It might be based on the prime numbers, for instance. * The individual digits of each term are manipulated to determine the next. * The sequence is based upon a constant, like pi or e. * It involves geometric / trigonometric ideas. * The sequence is not using the decimal system. * The sequence isn't purely mathematical - it makes reference to some independent external data or sequence. I list all these exotic possibilities for one reason: there will be a great many different rules that could produce such a succession of terms. Some of them will be more elegant than others, so it's a good idea to keep an open mind. But here is the more mundane hypothesis that I will work with: * The sequence is confusing because it is produced by a function that is neither linear or quadratic but of a higher degree. If you sketch a graph of the sequence*, it is absolutely clear that it cannot possibly be a quadratic function, because it is neither concave nor convex, but first one and then the other. What this means in mathematical terms is that the rate of change of gradient (the second differential) undergoes a sign change somewhere. For this to happen, d2y/dx2 must be variable, i.e. it is a function of degree 1 or greater. It follows that dy/dx must be at least degree 2, and the actual function must have at least degree 3. In other words, the function could be a cubic of form ax3 + bx2 + cx + d Normally cubic sequences can be identified at once because the third difference (the difference of the difference of the difference) is constant. This method requires 5 consecutive terms however, and we were only given three. To pin down the function that we are dealing with, we would need to solve for the coefficients a, b, c and d: four unknowns to find. It happens that we have been given 4 terms in the sequence, therefore we can form four equations and solve for four unknowns. Actually, that is why this approach is rather disappointing and lacking in elegance. If you are given a number n terms of a sequence, then it is always possible to find a valid rule using a polynomial of degree n-1. For instance, if you gave me the first twenty terms of the Fibonacci series, I could find a rule that worked using powers of x up to x19, but it wouldn't be the best answer. Still, I can't see a better way forward, so let's proceed with our cubic function. We have four simultaneous equations: a + b + c + d = 45 a + b + c + d = 45a + b + c + d = 39 8a + 4b + 2c + d = 39 a + b + c + d = 39 27a + 9b + 3c + d = 30 a + b + c + d = 24 216a + 36b + 6c + d = 24 The way to solve them is to rearrange an equation to get one variable in terms of all the others, and then to substitute for it in the remaining equations. Repeating this tedious process will eventually eliminate all but one variable: the solution for this one variable can then be used to find the value of the others. To cut a long story short, the solution is: a = 13/20 b = -108/20 c = 113/20 d = 882/20 We can write a function giving y in terms of x that applies to the sequence: y = [13x3 - 108x2 + 113x + 882] / 20 As predicted, it is far from pretty, and I suspect there was some simpler rule that you were meant to find. Still, this one is perfectly valid. It gives the missing terms as 21.9 andf 18.6, in that order. *Plotting the given values against their positions in the sequence, y against x.