SUCCESSIVE LIVES. 
189 
become q+t+qt^t+q (^ + 1) ; so that when the present value of the 
life in possession is the same as the present value of an annuity certain 
for t years, and the value of each of the q successive lives at the times 
of their respective nominations he the same as that of an annuity certain 
for ti years, we shall have the present value of all the lives, the same as 
that of an annuity certain for ¿ +?(A+ 1) years ; that is 
i 
Example. What is the present value of the next presentation to a 
living of the clear annual value of £500, supposing the age of the pre 
sent incumbent to be 65 years, the rate of interest 6 per cent, and that 
the age of the clerk at the time of presentation will he 28 years ? 
(Chester, Prob. Table 2.) 
i_i^(l + y 2 ) ¿=.06 a m ~a Ci —' r l.3751 ¥,=««=12.5987 
1+i 
7.3751 
.442506 
* 06 1 + Vs )= :557494X13.5987 = ,. 5gl 1Q 
1. 
l-ia m — .557494 
1+* 
1.06 
7.58119x 500=3790.595=£3790 11 11, 
the value required. 
PURCHASE OF ANNUITIES, &c. 
244. To find the annuity to be required on a single life for a certain 
amount of purchase money, so as to allow the purchaser a given rate of 
interest beside the premium necessary to secure his capital by a life 
assurance : 
Let s = the sum, 
i = annual interest of £l, 
p — annual premium for assurance of £l, 
a = the annuity. 
If we assume £l to be the sum advanced, and the annuity to be pay 
able at the end of the year, the last year’s interest must be assured in 
addition to the principal, viz. (l + i), the annual premium for which is 
p(l + 2>), which, subtracted from £l, leaves 
1—p(l-fi)=: the available principal, 
i -p(i-H) : f+pci+i): 
|s : « 
i+P (l+o 
1-K1+0 
= the annuity required.