Analytic Formula and Table of an increasing Life Anmiily . 1G7 



Respecting these numbers and tabulated values, which I have 

 extended to other rates, but cannot well augment here, it is to 

 be observed when at any time needed they are directed to be 

 calculated by M. De Moivre's Hypothesis, from which some neat 

 and appropriate formula has been investigated ; but it is ob- 

 vious the values deduced by it are only approximate to the true 

 ones as found from the series of decrements of life ; but this 

 latter value however can be obtained, and, excluding the hypo- 

 thesis, with greater ease, facilitv and accuracy of course, after 

 the manner ensuing: first, having from real observations and for 

 any order of increase determined the value of any such annuity 

 upon an advanced life, or beginning from an age near the end 

 of the table, tlien having for any form chosen so determined the 

 value at the age begun with, we may thence assign in succession 

 the value for every age intercepted, or that preceding, by aid of 

 a subsidiary formula ; and the operation being continued until 

 the Table is completed, as also similarly for single, joint or 

 longest of lives, the application may be generalized. 



Enumerating therefore the formula of derivation. Of the 

 order of numeral series in the table 1. 2. 3. 4. &c. 



Let A signify the value of 1/. annuity on life N, at age 7i — 1. 

 B, that at age n, but increasing by 1. 2., 3/. Then the value 



of such annuity for life N, is expressed by T A 4- 3 x 7^). 



If the annuity increases in the order 1. 3. 5. 7 ; if C be its 



c' 



value at age n, then is whole value for N = 2A -f- (C — 1). -^ = 

 If increasing by the triangular series 1. 3. 6. 10 ; and D be 



, . 1 f XT • , , (B + D)a' 



Its value at age n, the value tor i\ is = A -j . 



If it increases as the quadrangular series 1. 4. I). 16, the 

 value for N, = A 4- l—^ — —, 

 Where in each theorem ~ is the expectancy of 1/. on life N, a 



year hence. And, expanding by the combinatorial methods, other 

 forms of series will be developed, and the formula of continua- 

 tion applied. There are beside certain modifications fitting in 

 practice, and some relative to the deduction of value in the case 

 when an annuity is not for the whole term of duration, but tem- 

 porary, or in remainder dclerred ; but these arc not here ad- 

 mitting a full description. 



This being the first table of the value of a life annuity iii- 



crcasinc. 



