A L G E 
In this equation, any three of the four quantities, P, A, 
n, r, being given, the other may be found. 
' nA . . 
Cor. Let n be infinite, then P— -an infinite quan- 
z 
tity ; therefore, for a finite annuity to continue for ever, an 
infinite fum ought, according to this calculation, to be paid; 
a conclufion which Ihews the necellity of efiimating the va¬ 
lue of an annuity upon different principles. 
To find the amount of a given fum at compound filte¬ 
red. Let /?—tl. together with its intereff for a year; then 
at the end of the fu ll year R becomes the principal or fum 
due; therefore, 
i: R: : R: R -, the amou nt in two years; 
i : R:: R 2 : R 3 , the amount in three years, &c. 
in the fame manner R n is the amount in n years, and if P 
be the principal, the amount muff be P times as great, or 
PR"—M. 
„ M 
Cor. t. From this equation we have . 
Ex. What fum muff be paid down to receive 6ool. at 
the end of three years, allowing 5 per cent, per ann. com¬ 
pound intereff? 
In this cafe R— 1 .05, n~ 3, M— 600; and confequently 
_ M 600 , 
P = — =. = — t =518.302!. 
R A. 
following principle: “If the^prefent value of each pay¬ 
ment be determined feparately, the fum of fhefe values 
muff be the value of the w hole annuity.” 
Let a be the value or price paid down for the annuity, 
a the yearly payment, n the number of years for which it 
is to be paid, r the intereff of il for one year. The pre- 
fent value of the firft payment is —- the prefent value of 
tire fecond payment, or of a\, to be paid at the end of two 
a , . , r a , a 
years, is — ; -and loon; therefore a——— + 
“ 1.051 
Cor. 2. If P, R, and M, be given to find n, we have 
lo°‘. M — lost. P 
log. P+*X log. *=log. M, or»=- 2 —- jf—. 
To find the amount of an annuity in any number of 
years, at compound intereff. I.et A be the annuity, or 
fum due at the end of the firft year; then 1: R :: A : RA, 
its amount at the end of the fecond year ; therefore A-\- 
RA is the fum due at the end of the fecond year; in the 
fame manner 1: R :: 1 + 1 ? X A : R-\-R 2 x A, the amount of 
the two payments at the end of the third year ; and 
1 -}-/?+ R 2 X A is the whole fum due at the end of the third 
year; in the fame manner 1 +i?+ff 2 ... -j- R>‘ 1 X A is the 
R n —1 
fum due at the end of n years, that is, —- X A—M. 
Cor. In this equation any three quantities being given, 
the fourth may be found. 
To find the prefent value of an annuity to be paid for n 
years, allowing compound intereff. Let P be the prefent 
value, A the annuity ; then fince PR n is the amount of P 
R ”—1 
in % years, and ——~XA the amount of A in the fame 
R—i 
time; by the queftion, PR' 1 
R"- 
R- 
X A, and P — 
i-j-zr 
"i+r i+2r 
XA. 
M -1 
Cor. 1. Any three of the quantities, P, A, R, 71, being, 
given, the fourth may be found. 
Cor. 2. If the number of years be infinite, R n is infi- 
i . A 
nite, and —— vamflies, therefore P~— —-. 
R>‘ R—i 
Ex. If the annual rent of a freehold eftate be il. what 
is its value; allowing 5 per cent, compound intereff ? In 
this cafe A=zi, R — i=z.o$ ; therefore the prefent value 
P———— 20I. or 20 years purchafe. 
.05 
Cor. 3. The prefent value of an annuity, to commence 
at the expiration of p years, and to continue q years, is 
the difference between its prefent value for p-\-q years, 
and its prefent value for p years. 
Scholium. The method of determining the prefent 
value of an annuity at fimple intereff, as given above, has 
been decried by feveral eminent arithmeticians, and in its 
Head a folution of the queftion has been propofed upon the 
VO I,. I. No. 20. 
1 -f-rcr 
Thefe different conclufions arife from a circumftance 
which the opponents feem not to have attended to. Ac¬ 
cording to the former folution, no part of the intereff of 
the price paid down is employed in paying the annuity, till 
the principal is exhaufted. Whatever calculation be 
adopted, a is the fum out of which the feller can exactly 
pay the annuities according to the propofed mode of cal¬ 
culation. Let the annuity be always paid out of the princi¬ 
pal a-; then x, x — a, x —2 a, x —3 a, &c. are the fums in hand 
during the firff, fecond, third, fourth, &c. years, the filte¬ 
red arifmg from which is r;t, rx — ra, rx — ira, rx —3 ra, See.. 
that is, the whole intereff is nrx —1 + 2+3 • • • • n — 1 X ra, 
or nrx — n.- — -ra, which, together with the principal x y is 
2 
equal to the fum of all the annuities; therefore, 
n— r 
na-\-n. - ra 
- n —1 , z 
1 4 -nr.x—n. - rct—na, and x— ---. 
2 1 -\-nr 
According to the other calculation, part of the intereff,. 
as it arifes, is employed in paying the annuity, but not 
the whole. Thus the firft payment is made by a part of 
the principal, and the intereff of that part, which toge¬ 
ther amount to the annuity; and the other payments are 
made in the fame manner; this is, in effect, allowing in-' 
tereft upon that part of the w hole intereff which is incor¬ 
porated with the principal. According to either calcula¬ 
tion the feller has the advantage, fince the whole or a 
part of the intereff will remain at his difpofal till the laft 
annuity is paid off. 
If the whole intereff, as it arifes, be incorporated with 
the principal, and employed in paying the annuity, com¬ 
pound intereff is, in effect, allowed upon the whole. 
Let x be the price paid for the annuity, n the number of 
years for which it is granted, R, il. together with its inte - 
reft for one year. Then x in one year amounts to Rx, out 
of which tire annuity being paid, Rx — a is the fum in 
hand at the end of the firft year ; R 2 x—Ra is the amount 
of this fum at the end of the' fecond year, therefore 
R 2 x—Ra — a is the fum in hand at the end of tire fecond 
year; in the fame manner, R“x — R n 'a — R r ‘~ 2 a .... —a 
is the fum left after paying the laft annuity, which ought 
to be nothing; therefore R n xxx-R n 'a-\-RA 2 a .+a 
R n a—a R"a—a 
-, and w~- 
R- 
R n X li- 
On the Summation of Series. 
We have before feen the method of determining the 
fums of quantities in arithmetical and geometrical pro- 
greffion; butw'hen the terms increafe or decreafe accord¬ 
ing to other laws, different artifices muff be ufed, to obtain 
general expreffions for their fums. The methods chiefly 
adopted, and which may be confidcred as belonging to al¬ 
gebra, are, 1. The method of fubtraftion. 2.Thefum- 
mation of Recurring Series, by the fcale of relation. 3. 
The Differential Method. 4. The Method of -Increments. 
The inveftigation of feries whofe fums are known bv 
fi’.btractioH. 
3K 
Ex. 
