FINITE DIFFERENCES 409 



Thus if i* represents a term, in the middle of a set of given 

 values, M_ W (1 S)"M O will give the value of any term above u , 

 and the differences to be used must lie on the diagonal line running 

 upwards from U Q ; while u n = (1 + A "u will give the value of 

 any term below M O , and the differences to be used must lie on the 

 diagonal line running downwards from u . 



Example. The values 12, 12, 6, 0, 0, 12, 42 are seven con- 

 secutive terms of a series of which the number 6 is the 5th term. 

 Find the 1st term and the llth term. 



'-2 

 l -l 



Mo 



M 3 = 42 



If the middle value is denoted by U Q , the 1st term will be w_ 5 , 

 and M_ 5 = (1 - 8) 5 w = w - 5Sw + 108 2 M - 108 3 t< 



this relation ends at the 4th term, since S 4 w = 0, and taking 

 the differences on the diagonal line running upwards from u 

 Bu = - 6, S 2 w = 0, S 3 w = 6. 



Then w_ 5 = - 5 x ( - 6) + 10 x - 10 x 6 



= -80 



Also, the llth term is u 6 

 and w 6 = (1 + A) 5 w = w + 5Aw + lOA 2 ^ + 10A 3 w 



and taking the differences on the diagonal line running down- 

 wards from UQ, Aw = 0, A 2 ^ = 12, and A 3 w = 6 



Then W 5 = 0+5x0+10xl2+10x6 



= 180 



200. The results u n = (1 -j- A) n w and w_ n = (1 - S) n M will hold 

 for all values of n besides positive integers, and if the differences 

 in a certain column, say A r w, are zero ; exact results can be 

 obtained by simply taking the binomial expansion for (1 + A) n 

 or (1 8) n as far as the term involving & T ~ l u or S r - l u . Thus 

 if the value of w z . 4 was needed, it would be better to alter the 

 notation and call w 2 w , then M 2 . 4 could be taken as w . 4 and the 



