| a + nβ + | n·n − 1 | γ + ... |
| 1·2 |
where β, γ, ... are the first, second, ... differences of a; the coefficients being those of the terms in the expansion of (x + y)n.
4. Now suppose we treat the terms a, b, c, ... as being themselves the first differences of another series. Then, if the first term of this series is N, the subsequent terms are N + a, N + a + b, N + a + b + c, ...; i.e. the difference between the (n + 1)th term and the first term is the sum of the first n terms of the original series. The term N, in the diagram (A), will come above and to the left of a; and we see, by (ii.) of § 3, that the sum of the first n terms of the original series is
| ( | N + na + | n·n − 1 | β + ... | ) | − N = na + | n·n − 1 | β + | n·n − 1·n − 2 | γ + ... |
| 1·2 | 1·2 | 1 · 2 · 3 |
5. As an example, take the arithmetical series
a, a + p, a + 2p, ...
The first differences are p, p, p, ... and the differences of any higher order are zero. Hence, by (ii.) of § 3, the (n + 1)th term is a + np, and, by § 4, the sum of the first n terms is na + ½n(n − 1)p = ½n{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 (§ 14 (i.)) that all differences of a higher order are zero. Hence the sum of the first n terms is
| n + 7 | n·n − 1 | + 12 | n·n − 1·n − 2 | + 6 | n·n − 1·n − 2·n − 3 | = ¼n4 + ½n³ + ¼n² = {½n (n + 1)}². |
| 1·2 | 1·2·3 | 1·2·3·4 |
7. In § 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 § 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 §§ 9, 10 and 11.