Page:Elementary algebra (1896).djvu/438

420 Rh 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;