RECAPITULATION OF FORMULAS, 
59 
PRESENT VALUES OF SUMS AT COMPOUND INTEREST. 
82. Let p — the present value, s — the sum due, n — the number 
of years, i = the annual interest of £l. 
p = S (1 + i) n log p = — n log (1 +0 + log 9, 
s = p (l + i) n log s = n. log (1 + i) -f- log p, 
log s — log p 
n log (1 + i) 
83. When interest is convertible m times a year. 
1 + — ) log i? = — mn log ( 1 + + log 9, 
1 + ~ ) % * “ mn .log ( 1 + ~ ) + log p, 
log s — log J) 
c 
m.log ( 1 H—— 
m 
log s — log p 
m = — j-' 
n.log (l+± 
( 
AMOUNTS OF ANNUITIES AT COMPOUND INTEREST. 
84. Let s “ the amount, a “ the annuity, n = the number of 
years, and i ~ the annual interest of £l. 
When annuity and interest are payable once a year— 
(i + 0 n - 1 
■ J 
s = a. 
is 
a ~ 
(1 + i) n ~ 1’ 
log (I + 0 
{12 + (rc+l)j6}fl 
12 + 2 (n + 1) /3 
where /3 = ( — 'V.-i— l. 
PRESENT VALUE OF ANNUITIES AT COMPOUND INTEREST. 
85. Let p “ the present value, a — the annuity, n ~ the number 
of years, and i = interest of £l for one year. 
* For the investigation of this formula see Daily’s “ Doctrine of Interest and 
Annuities.”