CHAPTER XXIII 



197. If we consider the series 



1, 4, 10, 20, 85, 56 ... 



there does not seem to be any apparent connection between one 

 term and the term immediately following it, neither is it obvious 

 that the series is derived from some definite law of formation, 

 but if successive differences are taken, a study of these differences 

 will enable us to find the law of formation, providing such a 

 law does exist. 



Aw A 2 w A 3 w 



WD = 1 



3 

 % = 4 3 



6 1 



u z = 10 4 



10 1 



U 3 = 20 5 



15 1 



W 4 = 35 6 



21 



The values in the columns show the successive differences. 

 The first set being the result of subtracting each number from its 

 successor, the differences which are so obtained being changes 

 in the value of u, may be denoted by Aw. The second set of 

 differences are obtained by subtracting each value of AM from 

 its successor ; these values may be denoted by A 2 w, which may 

 be taken to represent " A operating on u twice " ; in the example 

 the values of A 2 w are in arithmetical progression ; therefore the 

 third differences A 3 w are constant, each being equal to 1, and so 

 all of the values of A 4 w will be zero. This indicates that the 

 series has been derived from some definite law of formation ; it 

 will be shown later that this law is 



u n = -(n 3 + 6n 2 + lln + 6) 



