Page:Elementary algebra (1896).djvu/441

The top layer consists of a single shot; the next contains 8; the next 6; the next 10, and so on, giving a series of the form

1, 1+2, 1+2+3, 1+2+3+4, $\ldots$ Series 1, 3, 6, 10, 1st order of differences 2, 3, 4, 2d order of differences 1, 1, 3d order of differences 0. Hence

529. Rectangular Pile. To find the number of shot arranged in a complete pile the base of which is a rectangle.

The top layer consists of a single row of shot. Suppose this row to contain m shot; then the next layer contains 2(m+1); the next 3(m + 2), and so on, giving a series of the form

$m,2m+2,3m+6,4m+12,\ldots$

1st order of differences $m+2,m+4,m+6$,

2d order of differences 2, 2,

3d order of differences 0.

Now let l and w be the number of shot in the length and width, respectively, of the base; then m $=l-w+1$.

Making these substitutions, we have

$S={\frac {n(n+1)(3l-w+1)}{6}}$.

EXAMPLES XLIV. b.

1. Find the eighth term and the sum of the first eight terms of the series 1, 8, 27, 64, 125, $\ldots$.

2. Find the tenth term and the sum of the first ten terms of the series 4, 11, 28, 55, 92, $\ldots$.