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2.1. Horner’s Method^55

show that

c

n ,.(k) _ ’

r=l

• h+l(n + l)(k+‘)

fork= 1,2,3 ,....
Since we know how to sum factorial powers, we can now sum ordinary
powers by expressing them first in terms of factorial powers. For example,
verify that r2 = d2) + r(l), and use this fact to derive the summation
formula for the first n squares.
Express r3 as a “polynomial” in factorial powers and use the result to
derive a formula for the sum of the first n cubes. Try out the process for
higher powers.
How can we systematically determine the factorial power expansion of
a given polynomial, such as r k7. Horner’s method can be adapted to this
purpose. For example, suppose r4 is to be written in the form

r4 = c,dn) + c,-ldnsl) + ... + clr f CO.

Observe that the polynomial r4 - cc is divisible by r; the polynomial r4 -
(cc + clr) is divisible by r - 1, and so on. We can use this to design a
“Homer’s” table whose entries are the desired coefficients.
Thus, a suitable table for r4 would be

1 0 0 0 0

0 0 0 0

1 0 0 0 0

1 1 1

1 1 1 1

2 6

1 3 7

3

1 6

Justify this table and read off the factorial expansion of r4 from it. Check
directly that the expansion is correct. Use this to derive a formula for the
sum of the first n fourth powers.