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