ALGEBRA. 
**5 
The film of n terms of the firft order is n ; therefore 
»+x is the increment of the fum of n terms of the fecond 
is the rath term of the 
n. 7 i- \-i . n-\-x 
is the n-\-1 til 
order, and its integral n 
third order; confequently, 
M 
term, or increment of the fum of n terms of the third or¬ 
der, and its integral- ^ is the fum of n terms of 
2-3 
the third order, or the sth term of the fourth order, See. 
-i.... n-\-m —i 
is the fum 
1.2 . 771 
the integrals requiring no cor- 
Thus it appears, that 
which was to be found ; 
r? ft ion. 
Ex. 7. Tq find the fum of n terms of the feries 1.24- 
2 - 3 + 3 - 4 + &c - 
Tlie a-j-xth term, or increment of the fum, is 
, r , , - , . 71 . n-l-1 . W +2 
and confequently the integral, or fum, is- 1 -; 
3 
which needs no correction. 
Ex. 8. To find the fum of the n firft terms of a feries 
whole Kth term is an 3 -\-brd fcnfd ; a, b, c, d , being given 
quantities. 
A flu me A.7i.7i-\- 1. n-\-z -\-B.n.n-\-i-\-C.nA-Dz=ea. 
+b . ; or 
A7l 1 -^-T,A7l*-{-2A7l-t-D' 
4 - Bn 2 -}- Bn 
-}- Cn 
an 3 f^an*-\-^anfa 
+ brd-\-2bn -\-b 
4- cn -j-c 
4 -d 
and, by equating the coefficients, A—a\ 2,AfB—T,a-\-f, 
or 3«4-ft—3«-W’» that is, B—b ; lAfBrfC—zaf-ibfC) 
or ia-\-b-\-C—2i a ~\~' L ^ J r c 7 hence Czzza-^-b-\-c ; alfo Dz=za-\- 
bfc-\-d ; therefore the increment of the fum is 
n-V-zfb .n.nfi -\-a-\-bfc.n-\-a-\-bfc-\-d, and the integral 
a.n — -i.n.v-\-i .n-\-x 
b. 71 —1 .n.77-I-1 
+-1 ~ b 
c-\-b-\-c. n —1 .n 
a-\-bfcfd.n\ which requires no correction. 
Though in general it is convenient to reduce an incre¬ 
ment to the products of arithmetical progreflionals, in or¬ 
der to obtain its integral; yet, if a quantity of any other 
form can be found, whofe increment coincides with the 
increment propofed, this quantity, when properly cor¬ 
rected, is the integral. 
Ex. 1. To find the fum of n terms of the feries 54-6-f- 
7 + &c. 
Let An-\-Bn*-\-Cn 3 -\- Sec. be the fum required; its in- 
crement is A.n-\-i-\-B.n+i) 2 -\-C. b+ 2 1 3 4 * &c. — An — 
Bn* — Cn 3 — &c. and the increment of the fum is alfo 224-5 > 
therefore, AfiBnfBffrffiCnfCf &c. =724-5 > and, 
by equating the coefficients, C=o; zB—i, or B—-i 
2 
A-\-B=$, or A =-; hence the fum required is —dk . 7 !. ; 
2 3 
which needs no correction. 
Ex. 2. To find the number of fliot in a pyramidal pile 
upon a fquare bafe whofe fide is known. 
Let n be the number in one fide of the bafe ; then a 2 is 
the number contained in the firft fquare; alfo, (ince one 
fbot in the next fquare will lie between every two of the 
former, n —i- is the num ber contained in the fide of the 
fecond fquare, and n —i| 2 the number in that fquare, &c. 
therefore the number of fhot is n 2 -}-^—1 K1.72—il 2 -}-. . . 
4"3 2 “|"i% or i 2 4 ” 2 M“3 a • • • ~\~n*. 
Suppofe ; 4 ?z 4 - 5 n 2 4 -Cn 3 tot e the fum of the feries; its 
increment is yf.ri 4 -i 4 -fi.' 7 + 7 ) 2 4 .C. «+1i 3 —., Brd — 
C« 3 ; and the increment of the feries is alfo «4'^ J » there- 
fore, 
3 c « s + 3 ^+C'l 
-\-2Biv-\-B '■•rr:)2 2 -{-2B4-i ; 
+AJ 
and, by equating the coefficients, sC=i, or C=- ; 3C4. 
3 
2B—2, or B—-) C-\-B-UA—i, or - 4-~4 -T=i hence 
2 3 - 
Az- 7 ; and the fum of the feries is - 4- — 4- — ; which 
6 ’ 623 
requires no correction. 
It An-\-Bn‘-\-Cn' , -\-Dn*-\- &c. be affumed for the fum of 
the feries, it is evident, from the procefs, that D and the 
coefficient of every fucceeding term vanilhes. 
Thofe who with to profecute this fubjeCt farther may 
confult Dr. Waring’s Fluxions, Sterling’s Summation of 
Series, and Emerfon’s Method of Increments. 
On Chances. 
If an event may take place in n different ways, and 
each of tliefe be equally likely to happen,- the probability 
that it will take place in a fpecilied way is properly repre- 
fented by -, certainty being reprefented by unity. Or, 
which is the fame thing, if the value of certainty be unity, 
the value of the expectation that the event will happen in a 
fpecified way is For the fum of all the probabilities is 
71 
certainty, or unity, becaufe the event muft take place in 
fome one of the ways, and the probabilities are equal; 
therefore each of them is -. 
n 
Cor. If the-value of certainty be a , the value of the 
expectation is But in the following articles we will 
71 
fuppofe the value of-certainty to be unity. 
If an event may happen in a ways, and fail in b ways, 
any of thefe being equally probable, the chance of its hap¬ 
pening is ——r, and the chance of its failing is--. The 
afb a-\-b 
chance of its happening muft, from the nature of the 
fuppofition, be to the chance of its failing, as a to b\ 
therefore the chance of its happening : chance of its hap¬ 
pening, together with the chance of its failing :: <2 : 
and the event muft either happen or fail, confequently the 
chance of' its happening together with the chance of 
its failing is certainty ; hence the chance of its happening: 
certainty :: a : a-\-b ; and the chance of its- happening 
=-- ^ - v . Alfo ftnee the chance of its happening together 
with the chance of its failing is certainty, which is rapre- 
■ a b 
fented by unity, 1- r , that is,-- is the chance of its 
a-\-b a-\-b 
failing. 
Ex. 1. The probability of throwing an ace with a lingla 
die in one trial is -; 
6 ’ 
the probability of- not throwing an ace 
is - ; the probability of throwing either an ace or a deuee 
(5 
• 2 o 
is -, arc. 
6 
Ex. 2. If n balls a, b, c, d t &c. be thrownpromifeuoufly 
into a bag, and a perfon draw out one of them, the proba¬ 
bility that it will be a is - j the probability that it will be-a- 
or b is 
n 
Ex. 3.- The fame fuppofition being made, if two balls 
b; 
