I 2 1 
for the Valuation of Life-annuities . 
In like manner, fuppofing money improved at an 
half-yearly intereft equal to half the yearly intereft, 
or to-, the value of the fame annuity payable half- 
been raoft convenient for me in thefc obfervations to make r ftand only for tlie 
intereft. In thefc writers, therefore, r fignifies the fame with i + r in this pa- 
per; andr — I the fame with r. 
It is faid above, that the value of an annuity payable yearly, on a life whofe 
complement is n } is - - 1 ■ q — ■ 2 — q — ”~~ 3 . , &c. continued to n terms. 
»Xi+r n X i-fTl «xi+rl J 
This expreflion is equal to — — q. - — ” ■ ,&c.(w) 
«xi+r «x i+H «xi+ri 
123 
+ ■==r i + 
n i+r 1 + rl" 1 + d 
,&c .(«). But 
+ 
n 
n 
Ac. (=— 1 - + * 
&c 0 =- 
«xi+f « X I + ;T « X I +r) 3 
—y (feep. 118.) Alfo, 
r r X 1 + 4 " 
7+rf i +/I 5 
by a theorem which will be demonftrated in the poftfcript, and putting 
a for any given quantity, -I q_ f q. -2-, &c. continued to n terms — : 
a 
a 
n 
X 
X 
a 
d—\X an a — 1 an a — 71 
. Therefore, if i+r is fubftituted for a. 
and v fori 1 the fum (multiplied by - ) of n terms of the feries 
r 'v m" n ' 
i+r 
+ 
X f+71 
+ J — , &c. will come out 
+ 7 T 7 TA 
nr 
Xy- 
1 
- X - 
r x 
7 R” 
or 
1 ^ } xy +y — 1 . Therefore, the feries - x —1— + 2 — q- ■■ , &c. con 
nr 
n i+r i+/T 7+7]* 
tinued to n terms and fubtrafted from the feries _ _ — }. ■ -1 — , Ac. con- 
1 +r 1 + 
r< 1 
+ 7 f 
tinued to n terms; that is, the value of the life will be y—— — Xy + v— 1 = 
nr 
1 1 +r 1 +r 
3 - xy — P xy — Y. 
r nr nr 
Vol, LXVII, 
R 
yearly 
