DISCOUNT AND PROtONGATION 1 1 



b. Discount, Prolongation and Rate. 



If $75 is put out at 4% compound interest for eight years, the 

 capital and interest to be paid at the end of the period, the $75 

 capital grows as follows : 



at the start, C(, = $75 ; at the end of 



first year, Ci^= 75 + interest for one year ^75 +75 (.04) 



= 75 (I +.04) ===75 (1.04); 

 second year, C2= 75 (1.04) + interest = 75 (1.04) (1.04) 



= 75(1.04=); 

 third year, C3 = 75 ( i . 04^) ( i . 04) = 75 ( i . 04=) ; 

 eighth year, 0^= 75 (1.04O (1-04) =75 (i-04'); 

 or, in general, C„ = Co fi.op") ; 

 that is: the final capital equals the initial capital multiplied by i.op". 



C„ 



From the above : = C„ or, the initial capital equals the final 



I. op" 



capital divided by i.op", or it is equal to the final capital discounted 

 for n years at p per cent. 



C„ 

 Also : — = I.op", or i.op" equals the final capital divided by the 

 C, 



initial capital. 



Since the numerical value of i.op" may be found in the table, 

 the value of p, or the interest rate is readily determined. 



Co (i.op") signifies that the initial capital Co is prolonged at p 



Cn 



per cent for n years. signifies that the final capital Cn is dis- 



i.op" 



counted at p per cent, n years. ^ 



c. Summation of Geometrical Seriez. 



If the yearly expenses on a property are $500, and these con- 

 tinue for fifty years and money is worth 3%, what will these ex- 

 penses amount to? Evidently the first $500 is out at compound 

 interest for 49 years, the second for 48 years, etc., and it is desira- 

 ble to find a short way of computing these various amounts. 



' The word capital is used here in a rather loose way, perhaps, but not 

 more so than in ordinary conversation, etc., and it is a very helpful term to 

 employ in these explanations. 



