I 2 6 
Dr. price’s Theorems 
By a like deduction, putting 
I 2 X 2 QX 3 4 X 4 r, »* 
c=- + — +^— ^+ — ’ &c. to ~ ’ 
<r « *r tf” 
. i 2x2x2 3x3x5 4x4x4 . n 3 
and D — - + — + y - + - — > 8ec. to — > it may be 
a a a s a* a n ] 
c K 1 t A + 2B+ I « + f " . J a + 3B + 3 c +I n+ 1 ] 3 
found that c = — — ; — > and d = 7 • 
a” + 1 a n a — 1 a n + i a n 
a — 1 
Andconfequently, fubftituting the values of a and b, that 
a a v 
C- 
2 an 1 
X 
a 1 -f a 
’“TV a-l a* ^Z7] z «» 
And, fubftituting the values of a, b, c, that 
a 2 + A.a z + a n 3 I 
D — \ a . - -X 
\an 
J=I) 4 
a n a — x a" 
-TV 
3 rt 2 » + 3an_ I 
" X . ■ . ■ 
< 7 3 +4 a 1 + a 
a — j| 
x ==p« Or, fince all but the firft terms in thefe expref- 
fions vanifli when n is infinite, that the fum of the feries 
&:c. continued infinitely is ^=j- 3 ; and that the 
fum of the feries ~ + “r+^T + ^; , &c. continued infinitely is 
a x -\~a 
o 2 -f - 4 & "4 q 
TT' 
Thele are all the theorems neceffary for calculating 
the values of annuities on lingle lives, and on any two or 
three joint lives, upon the hypothecs of an equal decre- 
ment of life. 
Suppofing r the interefl of jT . i for a year, the fum of 
n terms of the feries -7- + =--,+ =L=r;> 8ec. is the pre- 
1 + r 7+7) i+7V 1 
fent value of an annuity certain for n years; and 
1 
TTr 
