42 
ON THE VALUE OF ANNUITIES. 
i, and let z be its difference from the true value, then i = i' + z, and 
the above equation becomes 
pz 
(!+*' + z)~ n 
a a 
expanding the first side by the binomial theorem (Arith. and A Ig. 215.) 
, , w(w+1) 
z i 
(i + i'Y n — n (i -f i'Y (n+1) z + 
(1 ( ' ,+ 2) 2 2 — &C. 
pi' pz 
a a * 
Since z must be some very small quantity, the result will be very little 
affected, if we reject those terms in which the second and higher powers 
of z enter, which makes the equation 
By transposition, 
^ - n (1 + ¿ / )- (n+1) « = 1 - - (1 + i'Y :n , 
a a 
z || - n(i + ¿')- (n + 1) } = i - ^ - G 4- i'Y n ; 
dividing each side by - — n (1 + i'Y in+l \ 
Cl 
i _ YL _ (i + i'Y * 
a 
(n + 1) 
this being added to 1' will give an approximation to the value of i; and 
if upon trial it should not be found sufficiently correct, a value may be 
found still nearer by taking the value just found, and repeating the 
process. 
The Long Annuities, which have 30 years to run, are now sold at 19 
years’ purchase ; what rate of interest does the purchaser obtain for his 
money ? 
By Table 6, we find the rate lies between 3 and 3^ per cent. 
19.600441 = No. years’purchase at 3 per cent 
18.392045 ss ditto 3^ per cent 
1.208396 — difference 
.035 19.600441 
.03 19. 
As 1.208396 : 005 :: .600441 : .0024828 
Let i! = .03 + .0024828 = .0324828; 
then assume i = i' z = . 0324828 + z