304 
confequently the 
given in terms of m and z: 
root required, will be 
that 4s == 

z 378 3-5 75 
TT, 
and r= —— X r. 
i—*¢ 
But this feries may be inveftigated in a 
much fimpler way, by means of the bi- 
/ / 
nomial theorem only: for if 7, r, N, N, 
ty we Se the fame values as before, 
7 is to r, as 1-+y to 1—¥w, or yn ys (1 
+r)? :( Whence as 71=N, and 
7—N, we is —N(i-br), that is 
N—nN2. Py eegh Ee -N xz 
I—v7)>. 
AOE 30 a} 
Niv3, &c. —N-+aNa+ rn) ON xe + 
2 



LEAT, rie 
BEE ge eps ae, 
Aye. Z 
sada 
a 2 oe Moh SAU 7) —&o 
N-LN A: 
N—N_ oe 
. Therefore, eet m¢2—- 
a(N-LN) 
(1—1)(n—2) a (1--1)(2—2)(u+3) 
2-3 iN Aa 
I rc 
mis, &c.—=—.m, and by reverfion of fe- 
n 
ey 374—s nzte 
743 Sh aus 

res 4— 
373 - 579 
ees as before.— 
We thall illuftrate this rule hy a few 
examples. 
Example I. Let it be required to find 
the fquare root of 2. 
eu cast I 
Here N=2, 7=2, and ear -+ gin 
: / hs 
Sc: then if we aflumer==—, N is equal 
5 

ae el 
4 pe 525 
to oo) and 7 = = ar 
25 N+N 24+ 
25 
50—49 
Mathematical Correfpondence. 
mG —1) (x—2) 
[May 
—, &c.. and if the firft term of the fe- 
ries only~be taken for the value of x, r 
~ 


will be found = - oo ye a ee 
Bp ies, 2 54) )5G8 
2-99 
NES OS 
— = ——- = 1.4142 132 true to the laft §- 
Spee IS 
gure. But if the two firft terms be taken, 
cay! 205 5 
that is, if pm 30795 ant h & then 
8.993 7762392 
5461179 

will r = =1.41421356236 true 
38615055 4142135023 
to rr places of decimals. 
Example Lf. Required. the farfolid, or 
sth root of 125000. 
Here N=125000, m=5, the neareft 
/ / i 
root r=10, N=100000, and m= 
125000—I00000__ 25000 25 
Hare eae 



225000 
I 
== 
5:9 
8 
Therefore, —— m+ — m3, &e. = 
5 253 
3 : 2033 
shies er 
2033 
: 8 
2S a 10.456382, 
2033 59092 
QII25 
true to the laft place. 
Example III. Required the 365th root 
of 1.05. 

» andr = 




e 21 
Here N=r1.05=— 
20 
I 365*—1 
Therefore, +=— . 2-+———— 
RS 3.3653 
, and 7==36s. 
. 7:3, &e° 
el, ; f 
and if r==1, and confequently N=1, 2 
21 
—_—I 
20 

will be= 

mer =1.000136. 
Example 1V. Required the value of the 
2 
quantity 9)? 
f 
Here N=og, and n= : affume r=4, 
o—8 I 
l=3, 7.=—— =, and x= 
and N will =3, S18 Sey 
z. 
¥ 
