238 



5. This operation may be repeated until t is found sufficiently near. 



I will now take the total amount of interest that has to be paid on the 

 loan until it is all taken up. 



This on the first system will evidently be a t (V — 1), where v' is 1 + the 

 interest that has to be paid on one pound of the loan for a year. 



On the second system the interest payable at the end of the 



1st year would be (a— p) (v' — l) 

 2nd „ „ ( a -2p)(v'-l) 



3rd „ „ (a — 3p) (v' — l) 



(T-l)„ „ ( a -(T-l)p)(v'-l). 



But the year before the whole loan was taken up only— th of it would be left, 



it is evident that a— (T— 1) p=p 



So that we have an eqviidifferent series of which (a— p) (v'—l) is the first 

 term, p (v' — 1) the last, and T 1 the number of terms. The sum of them 

 therefore, or the whole interest to be paid on the loan 



!E=!{(a- P ) (v_i) +p( v_i) 



= a J V-l) (T _ 1) 



Therefore 



a (v'—l) . ( amount of interest ) . J amount of interest 



a (v — I) t . ^ I 1-1 ) ■■ J by first method J ' \ by second method 



Or 2t : T-l :: : . . . . 



But besides the interest on the loan there has also to be paid for the Sinking 

 Fund by the first method p t pounds, and by the second method p T pounds. 

 So that 



f amount paid for Sinking ) . ( amount paid for Sinking 

 P * • P -*- •• \ Fund by first method J ' \ Fund by second method 



And combining the two we get 



( whole amount paid ) . / whole amount paid 

 "■"P ' P ■-• \ by first method J ' \ by second method 



Or(p+2)t:(p+l)T-l :: : 



Now the limits of p are o and a, and as it gets small both T and t increase, 

 but t will increase slower that T for it also depends upon the value of v which 

 remains stationary. On the contrary as p gets large t will decrease more 

 slowly than T for the same reason, and the position of equality will of course 



depend upon the values of v and a. If however we take a > 1000 ; p< -r-., and 



v=1.05 — which in practice will include all cases — it will be firund that 



(p + 2)t<(p + l)T-l. 



The actual amount that would have to be spent by either method can be 

 easily found by substituting in the following formulae the different values for 

 a, p, v, and v'. 



