298 pbofessor kelland on general differentiation. 



JIquations of Differences. 



20. As the method of solving equations of differences of the second and 

 hio-her orders, by treating the symbols of operation as symbols of quantity, and 

 reducing the resulting fi-action by decomposing it into partial fractions, has been 

 little, if at all, employed, we shall commence with an example or two in ordinary 

 equations of differences. 



Ex. 1. «r + 3 + "•.r42 + *«.>- + l+'-'«,.; = X. 



This may be written 



{ (1 + a)^ + a (1 + a)2 + i (1 + a) + c }«<.,. = X. 



If we write a, for 1 + a, and suppose a, /3, 7 the roots of the equation 

 A,' + flA,- + 6A, + c=0; we get (A,-a)(A,-/3) (a,-7K=X, or 



'(,■= ^ .(X + 0). 



■' (A,-a)(A-^)(A-7) ^ ' 



This equation is reduced, by the decomposition of the fraction of operation 

 into its equivalent partial fractions, to 



+ 1 — — (X + 0). 



(7-«)(7-^) (A -7) 



-^Qyif _i — (X + 0) is the solution of the equation !;,.^i -a»,=X + 0; 



hence it is equal to a- (a+ s^tti) ^ and similarly of the others. 

 Hence the complete solution of the given equation is 



CoK. If a =|8, we must, as in similar cases, put a + c for ^, and expand in 

 terms of -■. The result is 



__(_+,a.-i-^(B + -«..i «. + . } 



-^7-(c + 2^ 

 (7-«)' V 7' 



a— 7\f a— 7 



