Find polynomial f(n) such that for all integers n >= 1 we have
f(n) = 1^3 + 2^3 + 3^3 + ... + n^3
Write f(n) as a polynomial with terms in descending order of n
+129850 Using the sum of differences
1 9 36 100 225 441 784
8 27 64 125 216 343
19 37 61 91 127
18 24 30 36
6 6 6
We have 4 non-zero differences
The polynomial will be of the form
an^4 + bn^3 + cn^2 + dn + e
We have this system
a + b + c + d + e = 1
16a + 8b + 4c + 2d + e = 9
81a + 27b+ 9c + 3d + e = 36
256a + 64b + 16c + 4d + e = 100
625a + 125b + 25c + 5d + e = 225
This is tedious to solve....using a little technology
a = 1/4
b = 1/2
c = 1/4
d,e = 0
The polynomial is
f(n) = (1/4)n^4 + (1/2)n^3 + (1/4)n^2
