Using the formula for the sum of n terms we obtain
526. It will be seen that this method of summation
will only succeed when the series is such that in forming
the orders of differences we eventually come to a series in
which all the terms are equal. This will always be the
case if the nth term of the series is a rational integral
function of x.
PILES OF SHOT AND SHELLS.
527. Square Pile. To find the number of shot arranged in a complete pyramid on a square base.
The top layer consists of a single shot; the next contains 4; the next 9, and so on to n^2, n being the number of layers: hence the form of the series is
.
Series .
1st order of differences 3, 5, 7,
2d order of differences 2, 2,
3d order of differences 0.
Substituting in Art. 525, we obtain
528. Triangular Pile. To find the number of shot arranged
in a complete pyramid the base of which is an equilateral triangle.
