1S8 
LIFE ASSURANCES. 
If x be equal to the value of an annuity on a life aged m x , the ex- 
pression will become 
■ îcl 
l+i 
- (1 + V 3 ) , which is the same thing as the 
present value of an assurance of (1 + Y 3 ) pounds on a life aged m x . 
From the nature of the reasoning in this article, it is evident that if 
V„ be the value of an annuity on the ?Rh life at the time of entering on 
possession, and the value of the (n—1) preceding lives be a’,..!, the 
present value of the rath life is 
1+2 
(1+V„). 
240. Let T n _!, the present value of the (n—1) successive lives, be 
equal to an annuity certain for the term of t years, and Y„ be equal to 
1_ r * 
the value of an annuity certain for t l years, then x n _ l = —— , and 
the expression 
1 —ix n -i 
1 + 2 
and the expression 1 + V n = 1 + 
1 — r tl (1+0-/1 
Q-f-Q-* 
1+2 ' 
; therefore, 
(1 + V.) = (1 +0- w,) x (1 - 0 'tin = 
1+2 2 
(1+2)-*— (l+Q- (Wl+1) _ r‘-r t+t ' +l 
i i * 
the present value of the nth life in succession; to which adding 
— / 
-7—, the present value of the (n—1) preceding lives, we have 
1 r ((+n+i) 
the present value of the n successive lives, which is the same as the 
value of an annuity certain for t + ^+l years. 
241. From which it appears that the effect of adding a life whose 
value at the time of nomination is the same as that of an annuity certain 
for the terra of t x years, is to extend the term of an annuity certain, 
whose value is equivalent to that of an annuity on all the previous lives 
in succession, by the term of ¿ t + 1 years. 
242. Also, that if a m be the value of an annuity on the life in pos 
session, and Y 2 , Y 3 , V 4 , &c., be the values of others that succeed it 
at the times of their respective nominations, while the terms of equiva 
lent annuities certain are t, t x , ¡f 2 , t 3 t q , the present value of all 
the lives in succession will be the value of an annuity certain for the 
term of (q-{-t+t l + t 2 +t 3 .., years. 
243. If each of the lives which succeed that now in possession be of 
the same value at the time of nomination, we shall have t l} t s , t 3 ,.. .+f i5 
equal to each other, and the expression (q+t+t x + t 2 + t z ... +t q ) will