na + 1/2n(n - 1)p =
1/2n{2a + (n - 1)p}.

6 As another example, take the series 1, 8, 27, ... the terms of which
are the cubes of 1, 2, 3, ... The first, second and third differences
of the first term are 7, 12 and 6, and it may be shown (S 14 (i.))
that all differences of a higher order are zero. Hence the sum of the
first n terms is

n.n - 1 n.n - 1.n - 2 n.n - 1.n - 2.n - 3
n + 7 ------- + 12 ------------- + 6 ------------------- =
1.2 1.2.3 1.2.3.4

1/4n^4 + 1/2n^3 + 1/4n^2 = {1/2n(n + 1)}^2.

7. In S 3 we have described b - a, c - 2b + a, ... as the first,
second, ... differences of a. This ascription of the differences to
particular terms of the series is quite arbitrary. If we read the
differences in the table of S 2 upwards to the right instead of
downwards to the right, we might describe e - d, e - 2d + c, ... as
the first, second, ... differences of e. On the other hand, the term
of greatest weight in c -2b + a, i.e. the term which has the
numerically greatest coefficient, is b, and therefore c - 2b + a might
properly be regarded as the second difference of b, and similarly e -
4d + 6c - 4b + a might be regarded as the fourth difference of c.
These three methods of regarding the differences lead to three
different systems of notation, which are described in SS 9, 10 and 11.

_Notation of Differences and Sums._

8. It is convenient to denote the terms a, b, c, ... of the series by
u0, u1, u2, u3, ... If we merely have the terms of the series, un may
be regarded as meaning the (n + 1)th term. Usually, however, the terms
are the values of a quantity u, which is a function of another
quantity x, and the values of x, to which a, b, c, ... correspond,
proceed by a constant difference h. If x0 and u0 are a pair of
corresponding values of x and u, and if any other value x0 + mh of x
and the corresponding value of u are denoted by xm and um, then the
terms of the series will be ... u_(n-2), u_(n-1), u_n, u_(n+1),
u_(n+2) ..., corresponding to values Of x denoted by ... x_(n-2),
x_(n-1), x_n, x_(n+1), x_(n+2)....

9. In the _advancing-difference notation_ u_(n+1) - u_n is denoted by
[Delta]un. The differences [Delta]u0, [Delta]u1, [Delta]u2 ... may
then be regarded as values of a function [Delta]u corresponding to
values of x proceeding by constant difference h; and therefore