96 BULLETIN 136, U. S. DEPARTMENT OF AGRICULTURE. 



payable at the end of each year, is designated by the symbol s^. To 

 find s»] each annual payment must be accumulated, at the effective 

 rate of interest i, to the end of the term of the annuity. The first 

 payment of 1 accumulates in n— 1 years to (l+i) n ~ 1 ; the second 



payment of 1, in n — 2 years, to (l+i) n ~ 2 ; etc ; the 



(n — l)th payment of 1, in 1 year, to (1 +i) ; and the nib. payment at 

 the end of the term is 1. Adding the separate amounts in reverse 

 order, there results 



s^ = l + (l+i) + (l+i) 2 + • +(l+i) n -K 



The sum of this geometric series is 



^= kiL± ^ — -• do) 



Values of this quantity are given for various rates of interest and 

 terms in Table 32. 



Example 5. — To find the accumulation in 47 years of an annual sinking fund of 

 1% of $1,000,000, if the fund is credited annually with 3% compound interest. 



This is an application of formula (10) where n=47 and i=.03; since s^7i=100. 3965009 

 the accumulation will be 



100.3965009 X$10,000=$l, 003,965.01. 



The same principles may be applied to find the amount of an 

 annuity for n years with annual rent 1 payable in p equal install- 

 ments during each year. The amount of such an annuity is desig- 

 nated by the symbol s^, and its value is represented by the- follow- 

 ing formula : 



If 1 4- % is replaced by (1 +j/m) m in accordance with formula (2), the 

 amount of the annuity is then expressed in terms of the nominal rate 

 of interest j with frequency of conversion m, thus: 



^ _ (l+j/m)"'-l {1Z) 



-m (l+j/m)~»-l , a) 



*■ p[0.+j/m) m ip-l] y J 



Example 6. — What will be the accumulation in 47 years of an annual sinking fund 

 of 1% of $1,000,000, paid in semiannually, if the fund is credited as received with 

 3% interest compounded annually? 



This is an application of formula (11) where 71=47, p=2, i=.03, hence 



(l+.03) 47 -l 3.0118950 

 S ^ = 2[(l + .03)*-l] = ~^^ = 101 - 143954 



and the accumulation will be 101.143954X$10,000=$1,011,439.54, 



