Partial Fraction Problem. 567 



Therefore 



*-- [ A 1 i*- i «-«>-(" + ?-r i ){rr i ) 



Nc 





and 



/a + £Va + £ + *-l\ /W£ + *-lV£ + *-l\ 



Hence 



**- fT<-<«--»>-rr i )r' 9 r->-«-.-~ 



Therefore, if £ is fixed, the assumed form of Q^ K is true 

 for all values of k. 



We shall next prove that the assumed form holds also for 



Q*+i, <• 



We may write 



IS*- 2 ' 3 / . Q\ 



or 



Q w = 2 X (-5)/i(-a)-f a + ^) 



/3 = 0a=0 \ P / 



» [ T""T<- w -*( T TT + 'r 1 ) 



X ?ft/c-2/3-a-2y-$« 



Letting a=/c—2/3—\, 



Q i+1 ,«= 2 2 (-by(-ay-»-*('-P-*) 



/3 = 0\=0 \ H / 



