3*3 
A L G E 
Ex.3. To'find the fumof ntermsof theferiesn—1 .» 
4-72—2 .x 2 -\-?i —3.* 3 4* _ _ 
In the feries n —1 . * 4 -n —2 .x--\-n 3 - x3 4 " ^ c .‘ t ^ e 
fcale of relation is 2- 1 ; therefore its lum in infinitum is 
n—i ■ x-H—2..r 2 —2.22— i.x z n- A f ter n 
terms the feries becomes — x n + 1 — ix n T 2 Sec. the him of 
xn 1 
which is found in the fame manner to be — -■; theiefore 
1 —*1 
n — 1 ,x-\-n —2 . a -2 -\-n —'3-^ 3 4~ ^ c ' to n terms, “ 
-i.-v n. — 2.x"- 
-- 4-1- &c. to 
n —i.x— nx^-\-x n ' 3 e l 
Hence 
•n terms, =- 
72—1 ,x — nx 2 4 -^:” + 1 
Ex. 4. To find the fum of n terms of the feries 
4 - 3 , x , 4 - 4 I, *' 3 "V" & c * 
Let the fcale of relation be f-\rg-\-b > then 
9 /+ 4 £+ 
16/+ 9 o -+4^=2 5 
25/4-i%4-9 / 5 = : 3 6 - . 
From thefe equations we obtain /=23, g— —3, 
h— 1, which values, when fubftituted, produce the 
fucceffi’Ve terms of the propofed feries; therefore S= 
r 4~4- v 49 xl 3 X r *y g 4 - 3 * z _.__!+!* ~h,p fnm of the fe- 
1— 3 * 4 - 3 **—i— x] 
ries in inf. w'hen x is lefs than 1. 
After the firft 2? terms the feries becomes 724-1')’* X *” 4 * 
x *” + *+i+ 3 > x *" + * 4 - &c. which arifes from 
the fraflionX 
_ i— 3 * 
—3 . Tz-f-il’x”' 1 '’ — 3 . n-\-2}- 3 ■ HL *x n + * 
-37c'— x‘ 
B R A, 
b—a-\-d* 
c~2b'-—a-\-d* * id* -\-d* * 
d—T,c —3^4a-f^‘ ** ~a-\-id* -\-T,d* * 4 "^* 1 * 
e—^d—Scff^b—a+d* v =ia+^d* 4 -6d* 4 4-4^ 4 4 -H 4 
See. & c. 
From which it is manifeft, that the coefficients of d, d*, 
d**,d***, Sec. in the expreflion for the 72-fitli term of the 
feries a, b , e, d, See. are the coefficients of the terms of a 
binomial raifed to the Ttth-power; that is, the 72-fith term 
of the feries is a-\-nd* -L-n.— —- d 11 4-77.-- d 111 4* 
2 23 
Sec. 
Cor. i. The 72th term of the feries is 224-72— id* 4 * 
n —1— dd* 4 4-?2—1.-——- d 1 * ‘ 4 - &c. 
2 23 
Ex. Required the 22th term of the feries, 1, 3, 5, 7, &c. 
t> 3 > 5 > 7 » 
222 
o o 
Herea=x, d* — 2, d** — o; therefore the 72th term is 
I + 7 ?- 1 . 2—277 - 1 . 
Cor. 2. If the differences at length vanifh, the 72th term 
of the feries will be exaftly determined ; but, if the diffe¬ 
rences do not vanifh, we can only approximate to it; and 
the lefs the differences become when compared with the 
former differences, and with 72, the nearer will the approxi¬ 
mation be to the true value of that term. 
Let the propofed feries be 0, a , a-\-b, a—J—<r, a-\-b-{-c 
4 -d, Sec. then 
iff Diff. a, b, c, d, Sec. 
2d Diff. b — a, c — b, d—c, Sec. 
3d Diff. c — 2 ^ 4 "®i d —2 c-\-b, Sec. 
4th Diff. d —304-3^— a, Sec. 
Sec. 
Let b — a—d* , c — ib-\-a—d* 4 , d — 3 c 4 " 3 ^— a—d* ! 1 , &c.. 
then the 724-1 th term of the feries, that is, a-\-b-\-c-\-d-\- 
and confe- 
H 4-f j C . X n 272^4-272 1 X X n -h'4*22 2 x"4 ' 
I- X 1’ 
quently the fum of n terms of the feries is 
* 4~*-724-lV.V ,? -|— 272*-}-272-1 X x”~F*- rfxv^g 
I-X | 3 
On the Differential Method. 
In any feries of quantities, a, b, c, d , e,f. Sec. if each 
term be taken from that which follows it, and the diffe¬ 
rences of thefe differences be taken, and fo on, the fol¬ 
lowing ranks of differences will be obtained: 
iff Diff. b — a, c — b,d — c, e — d,f—e, See. 
2d Diff. c — ib-\-a, d — zc-\~b, «— id-\-c,f—2e-\-d, Sec. 
3d Diff. d — 3 <: 4 ' 3 ^ — a > e ~ 3 ^ 4 \X— bif—ie+zd —c. Sec. 
4th Diff. e — i,d-\-6c —4^4-12, f—^t^-Sd —4c Sec. 
5th Diff./—5^4-10^—ioc4-56— a, 8e c. 
Sec. Sec. 
Hence it appears, that the coefficients of the quantities 
c, b, c, d, Sec. in the firft term of the 72th differences are the 
coefficients of the terms in the binomial raifed to the 72th 
power, and that their figns are alternately pofitive and ne¬ 
gative. 
Let d*, d**, d***, d iV , &c. reprefent the firft terms 
in the firft, fecond, third, fourth, See. orders of differen¬ 
ces; then 
d* —b z —a 
d * 4 —c — zb-\-a 
d * 11 —d —3C-|~3 b-—a 
d* w =e— 4 ^ 4 - 6 c— 4 ^ 4 -a 
&c. 
And, by tranfpofition, 
Vol. I. No. 20. 
&c. to n terms, is na+n. - a x -•—— d x 1 -|- 
2 23 
If therefore a, b, c , d. Sec. be the terms of any feries, 
whofe firft, fecond, third, Sec. differences are reprefented, 
by d*, d**, d lii , Sec. the fum of n terms of this feries iz> 
na-\-n. 
-±d*+n.~ 
2 2 
- d 4 4 4- Sec. 
Ex. 1. Required the fum of n terms of the feries i 4 s 
+ 5 + 74 - &c - 
J> 3 » 57 7 » 
2, 2, 2, 
o o 
In this cafe a— 21, d*—z, d**~ 0; hence the fum is 
724*22.72—1 =72”. 
Ex. 2. Required the fum of 72 terms of the feries r-f-j* 
+ 3 ° 4 - 4 a -i- 
1, 4, 9, 16 
3 > 5 * 7 
2 , 2 
O 
Here a~i, rf 1 =3, d** — i y d* li — o; therefore the fum 
77-1 , 77-1 72-2 
— 724 - 72 .- X 34 -”--•- X 2 : 
72.27? 2 4-37z4-I 
72 . 72 4 - 1 . 2724-1 
1.2.3 
Ex. 3. To find the fum of the feries ! 5 4 -A 4 - 3 3 4 - 
to ?2 terms. 
1, 8, 27, 64, 125 
7 ; t 9 > 37 . 61 
12, 18, 24 
6 , 6 - 
o 
3 L 
Here 
