what's the next number in this sequence? -14409, -6483, -2503, -771, -165, -19, -3, …?

Question

how did you found that?

Answer 1

The first thing I did was to test whether this sequence might be generated by a polynomial. I did this by computing the differences at each term, and seeing whether I got any rows where any differences are the same. Note that it is always possible to find a polynomial of degree n-1 that passes through n numbers, so the main reason to do this test is to ascertain whether any polynomial of smaller degree will suffice. If one does, then the sequence is likely generated by that polynomial. If it doesn't, then the actual generating function of the sequence is probably something else, and unrelated to the polynomial that passes through these points.

The rows of differences are:

-14409 . -6483 . -2503 . -771 . -165 . -19 .... -3
........7926 .. 3980 ... 732 ... 606 ... 146 ... 16
.............-3946 .. -2248 . -1126 . -460 . -130
.......... .......1698 ..... 1122 ... 666 .. 330
.......... ......... .......576 ..... 456 ... 336
.......... ......... .............120 .... 120

As you can see, the fifth row of differences is all the same (counting the top row as zero), and thus there is a polynomial of degree less than 6 that runs through all the points. Thus, these points were probably generated by the polynomial.

We will now find the polynomial. Since the polynomial is of degree 5, it satifies:
Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex + F = y

We thus generate the following system of equations:

A(1^5) + B(1^4) + C(1^3) + D(1^2) + E(1) + F = -14409
A(2^5) + B(2^4) + C(2^3) + D(2^2) + E(2) + F = -6483
A(3^5) + B(3^4) + C(3^3) + D(3^2) + E(3) + F = -2503
A(4^5) + B(4^4) + C(4^3) + D(4^2) + E(4) + F = -771
A(5^5) + B(5^4) + C(5^3) + D(5^2) + E(5) + F = -165
A(6^5) + B(6^4) + C(6^3) + D(6^2) + E(6) + F = -19
A(7^5) + B(7^4) + C(7^3) + D(7^2) + E(7) + F = -3

This results in the following augmented matrix, which we can solve by gauss-jordan elimination:

[1, 1, 1, 1, 1, 1 | -14409]
[32, 16, 8, 4, 2, 1 | -6483]
[243, 81, 27, 9, 3, 1 | -2503]
[1024, 256, 64, 16, 4, 1 | -771]
[3125, 625, 125, 25, 5, 1 | -165]
[7776, 1296, 216, 36, 6, 1 | -19]
[16807, 2401, 343, 49, 7, 1|-3]

I'll spare you the tedious steps of solving and get straight to the solution:

[1, 0, 0, 0, 0, 0 | 1]
[0, 1, 0, 0, 0, 0 | -39]
[0, 0, 1, 0, 0, 0 | 608]
[0, 0, 0, 1, 0, 0 | -4736]
[0, 0, 0, 0, 1, 0 | 18432]
[0, 0, 0, 0, 0, 1 | -28675]
[0, 0, 0, 0, 0, 0 | 0]

Thus, A=1, B=-39, C=608, D=-4736, E=18432, and F=-28675. Thus the generating polynomial is:

f(x) = x^5 - 39x^4 + 608x^3 - 4736x^2 + 18432x - 28675

And the eighth term in the sequence is -3 (as is the seventh term). The first 20 terms are:

- 14409, - 6483, - 2503, - 771, - 165, - 19, - 3, - 3, - 1, 45, 321, 1277, 3747, 9069, 19205, 36861, 65607, 109997, 175689, 269565

Edit: EamsMan, I believe that you transposed the four and the eight in the second term when entering the values into Excel. As a result, your polynomial is incorrect.

Answer 2

1

Answer 3

To the previous poster: well, yeh, with 7 numbers given there will be a polynomial of degree six or less that generates the sequence. I doubt that is the intended solution (although I am stumped on what the intended solution really is). :)

Answer 4

2517