Page:Elementary algebra (1896).djvu/438

By addition, since we obtain . But

Hence if the law of formation holds for $u_{n+1}$, it also holds for $u_{n+2}$, but it is true in the case of $u_{4}$, therefore it holds for $u_{5}$, and therefore universally. Hence

If we take a as the first term of a given series, $d_{1},d_{2},d_{3},\ldots$ as the first terms of the successive orders of differences, any term of the given series is obtained from the formula

.

525. The Sum of n Terms of the Series. Suppose the series $u_{1},u_{2},u_{3},\ldots$ is the first order of differences of the series

$v_{1},v_{2},v_{3},v_{4},\ldots$

then identically ; Hence in the series the law of formation is the same as in the preceding article;