ON THE VALUE OF ANNUITIES. 
assume i r as a quantity somewhat near the true value of i, and let the 
true value be i = i' + z; then the equation becomes 
(! + *' + 2) 
l ' + - 2, 
expanding (I+z'+2)“ d by the binomial theorem (Arith. and 
Alg. 275.) we have 
d {d + 1) 
(1 + i')~ d - ¿(1 +«')- (d + ,) 2 + 
■(1 -f ¿')- (d+2) 2 2 —, &C. 
v}!_ + Pz. 
a a 
rejecting the second and higher powers of z, which being very small, 
will not much affect the result. 
(1 + i')~ d - d (1 +0“ cd+1) 2 = V — + 
by transposition, cZ (1 + (d + 0 z + z — (1 + i') d 
pi 
7 
a