Page:Elementary algebra (1896).djvu/440

422 Rh 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

$$1^2, 2^2, 3^2, 4^2, \ldots, n^2$$.

Series $$1, 4, 9, 16, \ldots, n^2$$.

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.