ALGEBRA. 43 



and the like law for q. The solution of this equation, a n , 5, being 

 functions of n, must involve two constants, since it is necessary 

 that two terms be known in order to determine the remaining 

 terms by this formula. These constants may conveniently be 



taken p u p t ' q ly q t respectively. The fraction -- thus determined 



will not usually be in its lowest terms. We will take as an 

 example the question "To find the w th convergent to the continued 

 fraction 



1 1 4 12 2n(n-I) 



1_ 3-6- 9 - ...- 3n -.... 



Take u n to represent eitlitr p n or q n (since the same law holds 



>th), 



thru u n+l = 3nu n -2n(n- 



or 11^ - 2nu n = n{u n - 2(n - 



So "*-2(>i-l)u n =(n-l){u._ l -2(n-2)u m _ t }, 





u 3 - 4w, = 2(u t - 2w,), ( m 2 or as u = p or q). 

 1 1 t-nce w w+1 2nu m = Jn or 0, 







12 



and the n lk convergent is 



r-i 



r ' ' 



