C 73 ] 
Wherefore, the value of an annuity of i /. granted 
for the time s, will be expreffed by the feries 
a-\-b ( sa — 2 a-\-zb , sa — 3 «-|— 3 ^ ( sa — 4 a-j-^b 
ir * sar z * sar 3 * sar 4 
&c. to -f* — > this feries is divifible into two other 
1 ar* 
feries s, viz. 
4 4 -+^ + -^ + 44 &c.to+ 
s — s 
sr s 
2 d. 
t x JL + JL -f- i_ 4- ±y & C - t° _i_. 
a sr sr 1 sr 3 ,»- 4 sr 5 
Now, lince the firft of thefe feries’s begins with a 
term whofe numerator is s — i, and the fubfequent 
numerators each decreafe by unity ; it follows, that 
the laft term will be — o ; and, confequently, that 
feries expreffes the value of a life necelfarily to be 
extind in the time s. The fum of this feries may 
be efteem’d as a given quantity j and is what I have 
exprelfed by the fymbol in problem i. 
The fecond feries is the difference between the 
two following feries’s. 
- X't'T-3 + T"f^7 + &?• to 
a 
b 
- X 
a 
— 4 - s - — 7 4 “ ~jr s -T & c ‘ ro 4 - - . 
sr ' sr sr* sr * sr s • 
Where, negleding the common multiplier t 
the firfl feries is the value of an annuity certain 
to continue s years 5 which every mathematician 
knows how to calculate, or is had from tables al- 
ready compofed for that purpofe : this value is what 
I have called IP $ and the fecond feries is JL 
K Therefore 
