Dr. price's Theorems 
•ii 8 
But to prevent all perplexity on this 1 object, I will iiih- 
join the following inveftigations, which will be eafily un- 
■derifood by thofe who are acquainted with the common 
methods of calculating the values of life-annuities. 
Let r, as before, be the intereif of £. i for a year. 
Then the prefent value of £. i payable at the end of one 
year, two years, three years, Sec. will be 
i 
i+r 
Sec. refpebtively. And the prefent value of an annuity 
certain for n years payable yearly is the fum of this ie- 
, iii 
vies continued to n terms w, or -- — =r = p -—-y. 
In like manner, the prefent value of half jC. i (that is of 
i o s.=£. 0,5) payable at the end of half a year, a year, ayear 
and a half, Sec. reckoning half-yearly interefh at half the 
annual intereif, is 
0,<J t» > Sec. And the pre- 
q TV T 7 ! 
2 2 ' 2 * 
fent value of an annuity certain payable half-yearly for 
n years, each payment to be half the yearly payment, 
is the fum of this feries continued to in terms; or, 
o>5 
o,5 
r r r 
-X 1 +- 
2 2 2 
I 
r 
r X I + - 
2 
in 
% 
•=p- 
r x 1 + - 
-b. 
(l>) In the poftfcript it will be proved, that the fum of n terms of the feries 
A 4. _L _l _L 4- _L , &c. is 1 Subftitute i + r for a, and it will 
0 ^ a— 1 a* X a — I 
appear, that the fum of n terms of the feries — [. -- 1 — _[ — J — , &c. is 
I + r i +rf 1 +d’ 
1 1 
rx 1 
r 
By 
