# Thread: Formula for given sequence

1. ## Formula for given sequence

How would you find a general formula for a sequence:

0, 1, 4, 10, 20, 35, 56, 84, 120

And by general formula I don't mean P_n = P_{n-1} + P_{n-2} or anything like that.

Thx

2. Expressed as a combinations formula
C(n+2,3)

That is,
(n+2)(n+1)(n)/6

3. Originally Posted by DivideBy0
How would you find a general formula for a sequence:

0, 1, 4, 10, 20, 35, 56, 84, 120

And by general formula I don't mean P_n = P_{n-1} + P_{n-2} or anything like that.

Thx
Hello,

xamine the given sequence: ri means ith row
Code:
n:  1    2    3    4   5    6    7    8    9
-----------------------------------------------
r1: 0    1    4   10   20   35   56   84   120
r2:   1    3    6   10   15   21   28   36
r3:      2    3    4    5    6    7    8
In r1 is the given sequence, in r2 are the differences between the 2 numbers of the sequence which are above the difference, in r3 are the differences of the differences.

As you easily can see in r3 is an arithmetic sequence. Therefore the given sequence is an arithmetic sequence of 3rd degree.

The general equation of such a sequence is:

s_n = a*n³ + b*n² + c*n + d

You have to know the coefficients a, b, c, d

You know:
s_1 = 0 = a + b + c + d
s_2 = 1 = 8a + 4b + 2c + d
s_3 = 4 = 27a + 9b + 3c + d
s_4 = 10 = 64a + 16b + 4c + d

Solve this system of simultanous equations. Easiest method here is elimination. You should get: a = 1/6, b = 0, c = -1/6, d = 0

Thus the equation of this sequence is:

s_n = 1/6*n^3 - 1/6*n

4. ## Why does the method work?

Could you let me know why the method you presented works? By method I mean showing the rows and differences, etc. until you reach an arithmetic sequence. Could you suggests a reference for additional reading? Thanks

5. Two things:

1) A basic background in differential calculus (likely a first introductory course) will suggest it to you. If you are not ready for that, it will have to wait.

2) These are not reasonable problems. They are just barely mathematics. Mostly, they are just parlor games. There is something to be said for pattern recognition, but even then there is little to be learned. In EACH case, given a finite number of values, there are INFINITELY many solutions for the 'next' value. Some will argue for a "simplest" solution, but that is not ever what the problem statement says and it is impossible to define. Some will argue for a "logical" extension, but that is no more clear. If you have a next element, and can support its development, your working should be given full credit.