397 



APPENDIX VI. 



Notes on the Formulae for Compound Interest (see page 116). 



Summation of Geometrical Series. 



Let a be the first position in a geometrical series, q the factor 

 with which the first position is multiplied in order to obtain the 

 second position, and so on, and 11 the number of positions. Then 

 the sum 



S = a + aq + aq" + aq^ + . . . + aq"~K 



By multiplying both sides with q, the equation becomes 

 Sq = aq + ags -|- aq^ + aq''' . . . + aq''\ 



Deduct the first from the second equation : 

 Sq - S = aq" - a, or 

 S{q -l) = a ir - 1). 

 S = _i£ — Z — I (for a rising series). 



This is the formula for the sum of a rising series. In the case of a 

 falling series (g < 1), a more convenient form is obtained, by 

 multiplying the formula above and below by - 1, which gives the 

 formula — 



S = — ^ — Z_^' (for a falling series). 



If now n = CO , then q'^ = 0, and 

 S = (for a falling infinite series). 



Formula III.—C„,,, = ^ O-'^P""' - ^) is thus obtained : 

 C',„„ = E + B X 1-Op'" + B X 1-Op^"' + . . . + B X VOf^-''"\ 



