DISCOUNT AND PROLONGATION I E 



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, C\ 75 -J-- interest for one year = 75 +75 (.04) 



=^75 (i+. 04) ^75 (1.04); 

 second year, C 2 = 75 (1.04) -{- interest = 75 (1.04) (1.04) 



= 75(i.o 4 2 ); 



third year, C 3 = 75 (1.04*) (1.04) = 75 (i-O4 3 ) ; 

 eighth year, C 8 = 75 (i.O4 7 ) (1.04) 75 (i.O4 8 ) ; 

 or, in general, C n = C ( I . op n ) ; 

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



c n 



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



i.op n 



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

 for n years at p per cent. 



G. 



Also : = i.op n , or i.op n equals the final capital divided by the 

 Co 



initial capital. 



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

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



C (i.op n ) signifies that the initial capital C is prolonged at p 



Cn 



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

 i.op n 



counted at p per cent, n years. 2 



c. Summation of Geometrical Series. 



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. 



2 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. 



