3 42 
Mr. Morgan 
SOLUTION, 
Let r denote / i. increafed by its interefl: for a year, and let 
all the other fymbols be the fame as in the preceding problem. 
Let the life of B alio be fuppofed to be the older of the two 
lives; and then it will follow, by real'oning as in the folution 
of that problem, that the prefent value of S to be received on 
death of A, ihould that happen in the life-time of B, will be ex- 
b + c . a' c + d . a" d+e . a 
2abr 
+ 
lab i ’ 
+ 
labr • 
+ 
’+/. a 
prefled by the feries S x 
&c. This feries may be refolved into the two following ; 
lab t 
S ca' da n ea 
- X — - + - 7 — 4* 
2 abr aor 
/// 
/*' 
/>// 
i /" & c . . S 6a' ca" da"' fa 
abr 3 abr 0 ' 2 abr ah'/ 1 abr 2 abr 4 
&C. 
The flrfl of tnefe two feries may be again refolved into 
S c ca — ca' d da — da' — da" e 
„ x 7 : r — ~ — r — 
ea — ea —ea" -- ea"' 
br 
abr 
br 2 
abr 
br 3 
abr^> 
&C. 
. S da' ea' ea n \ o 
{ x — — — &c. — ) — S x 
V 2 abr % abr 3 ' 
c d da — da' e 
X ■■ d ; — 
cr acr cr 
2 br 
ea — ea 
■ea 
acr 
.2 
&c. Let B denote the value of an annuity on the 
life of B, C the value of an annuity on a life one year older 
than B, AB and AC the values of annuities on the joint lives 
of A and B and of A and C, and thefe feries will be = 
SxB — AI3 Sx fxC-AC 
2 zbr 
Again, the fecond feries above men- 
tioned, or x + — 3 &c., by purfuing the fame fleps 
£xS 
may be found = Llf x K — AK - — — where (3 denotes the 
number of perfons living at the age of a perfon one year 
younger than B, K the value of an annuity on that life, 
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and 
