452 Mr. Adams on Compound Interest. (Dec. 
a 
ay aid 
: Rar amount of P nm years since, or the 
present value of P due at the end of n years. 
Art. 4.—If the payments of interest be supposed to be made 
ite “aie 
>=" will give 
at m equal intervals in a year, then will 
. : Test 
~ m 
‘Art.2. If P = 1l.and m infinitely great, then will 
1 n? r? n3 r3 ni rt 
1 
( pase bn tea A ee Loe ee 
1+ 7) ! 
™m 
= anumber corresponding to the commonlog. — nr x *43429448, 
&e. 
Art. 5.—If the principal P, instead of increasing as in Art. 1, 
decrease.in the same ratio, we shall have, by a like manner of 
reasoning, P (1 _ =)" = P (1 — ry" =ss, the decrease of P 
" P 
in n years, and TE) is s! = present value of P at the end of 
n years. Hence it appears, that when the principal is continu- 
ally increased by the addition of the interest, the present value 
of P to be received nm years hence will of consequence be /ess 
than P; so, on the contrary, when the principal is continually 
diminished by subtracting the interest, the present value of P, to 
be received 7 years hence, will be greater than P. 
Art. 6.—Ifin Art. 1, m P = s, then will P (1 + 7” = mP; 
therefore (1 + 7)" = m. 
Art. 7-—So likewise in Art.5. 1f~ = s, then will P (l—r)" 
¢ mm 
P ye 2! 
= —; therefore (1—7)" = =. 
Art. 8.—To find the amount of 11. payable every v years dur- 
ing n years, the interest payable yearly. 
_ Let 1/. represent the sum payable every v years. 
(1 + r)’ = amount at the end of the first period, 
(1. + r)” = amount at the end of the second period, 
(1 + 7)” = amount at the end of the third period, 
1 + r)"-° = amount at the end of the v periods, 
therefore, 1+ (1 +7) + (1+ ryf°4+(1 +7)" 4+....0.... 
d4¢n = ws = amount of 1/. payable every v years. 
during m years.....+. aicyece ay ele (a) 
Art. 9.—When v = 1, equation (a) will become pbs glia 5 
amount of an annuity of 1/. per annum for » years, and a x 
ai+r)"—1 : - 
—— = amount of an annuity a@ for years. 
