FINITE DIFFERENCES 405 



This result enables us to find the value of any term, providing 

 the first term u is known and also the successive differences 

 A//,,, A 2 w , A 3 M . . . are known. These differences evidently lie 

 on a diagonal line running downwards from U Q . 



Symbolically, then, 



!=(! + A)w = (1 + A)M O 



M 2 = (1 + 2A + A 2 K = (1 + A)^ 



w 3 = (1 + 8A + 3A 2 + A 3 )w = (1 + A) 3 w 



4 = (1 + 4A + 6A 2 + 4A 3 + A 4 )w = (1 + A)X 



u 5 = (1 + 5A + 10A 2 + 10A 3 + 5A 4 + A 5 )w = (1 + A) 5 w 



, n(n 1) . n(n l)(n 2) . 

 andti n = (l + rcA+ v ; A 2 + - r^ ^A 3 . . .) w = (1 + A) n w 



In which (1 + A) 2 w means that (1 4- A) operates on u twice, 

 or more generally (1 + A)"w means that (1 + A) operates n times 

 on U Q . 



Example 1. Find the 9th term and the general term of the series 

 1, 4, 10, 20, 35, 56 ... 



Aw Ahi A 3 w A 4 w 



The quantities in the diagonal line running downwards from w 

 will give the values of u , Aw , A 2 w , A 3 w , etc., for this particular 

 series, and u = 1, AM () = 3, A 2 w = 3, A^ = 1, and A 4 M = 0. 



The 9th term is evidently u 8 



and w 8 = (1 + A) 8 M 



= w + SAw,, + 28A2MO + 56A 3 w 

 The relation ends at the fourth term since A 4 ^ - 0. 

 Then W 8 =l + 8x3+28x3+56xl 



- 165 



