96 BULLETIN 136, U. S. DEPARTMENT OF AGRICULTURE. 
payable at the end of each year, is designated by the symbol sz. To 
find sj each annual payment must be accumulated, at the effective 
rate of interest 7, to the end of the term of the annuity. The first 
payment of 1 accumulates in n—1 years to (1+72)"—'; the second 
payment ‘of 1,.im’ n—2 years, to (1 --1)"—2 Wetec.) s ee 
(n—1)th payment of 1,in 1 year, to (1+72); and the nth payment at 
the end of the term is 1. Adding the separate amounts in reverse 
order, there results 
eA ie) ee ca)? ahh be Sn ha oe tc + (1 +4)" =% 
The sum of this geometric series is 
(1+4)r—1. 
Sn] — B 
(10) 
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 1=.03; since sz77=100.3965009 
the accumulation will be 
100.3965009 & $10,000 =$1 003,965.01. 
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 sY, and its value is represented by the follow- 
ing formula: 
Gy wees 
Sn pl +e) 1 a 
If 1+7 is replaced by (1+7/m)™ in accordance with formula (2), the 
amount of the annuity is then expressed in terms of the nominal rate 
of interest 7 with frequency of conversion m, thus: 
oe (1 + 9/m)m sae | 
A CRgaor I o 
O (1 +9 /m)nr — I 
Su pl + J/myn? — 1 a 
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 n=47, p=2, i=.03, hence 
(1+.03)*7—1 3.0118950 
(2) ———._ _. — . 
“27l = 9 @ ae osyhan) an NOOO Taman. ean a 
and the accumulation will be 101.143954x$10,000=$1,011,439.54, 
