404 



PRACTICAL MATHEMATICS 



198. Let a certain series be denoted by u , u lt u 2 , u 3 , etc. 



Aw + AM O + AX 



= U Q + 2Aw + AX .... .......... (5 



= (M O + 2AM + A 2 M ) + (AM + 2A 2 M + AX) 

 = M O + 3AM + 3A 2 M + AX ...... 



M = M 



3A 2 M + 3AX 



= (M O + 3AM + 3A 2 M + AX) + (A 



+ AX) 

 = M + 4At* + 6A 2 M + 4AX + AX ...... 



M 5 = M 4 + AM 4 



= (M O + 4AM + 6A 2 M + 4 A 3 M + AX) + (AM + 4AX 



6A 



10A 2 M + 10AX 



A 



The multipliers of the differences are evidently the same as the 

 Binomial coefficients in the expansions for which the powers 

 1, 2, 3, 4, and 5 respectively. 



Hence 



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

 . = M + wAM + . ; A 2 M + -i r^ ^AX + etc. (( 



t .a o 



u 



