ALGEBRA. 
2 93 
Ex. 3. If the firft term of an arithmetical progrefiion 
be 14, and tire futn of S terms zZ, what is the common 
difference ? 
~~~ n 
Since za-\-n — i.by.-—s 
2 a-\-n- 
— 
n 
2 s 
Therefore b 
n —1 . bz=- 
n 
zan 
-za— - 
zs- 
n. 71- 
ii 4, n— rS; 
therefore, b~- > 
In the cafe propofed, s—z 8, 
7—28 
-224 
8X7 7 
Hence" the feries is, 14, it, 8', 5, Sec. 
On Geometrical ProgreJJion. 
Quantities are faid to be in geometrical progreJJion y or con¬ 
tinual proportion, when the firft is to the fecond as the 
fecond to the third, and as the third to the fourth, &c. 
that is, when every fucceeding term is a certain multiple, 
or part, of the preceding. If a be the firft term, ar the 
fecond, the feries will be, a, ar, aP, ar 1 , ar % See. For 
a : ar :: ar : ar 2 :: at* : ar 3 , Sec. 
The conflant multiplier is called the commmon ratio, and 
it may be found by dividing the fecond term by the fidt> 
If the two extremes, and the number of terms in ageo- 
metrical progrefiion be given, the means may be found. 
Let a and b be the extremes, n the number of terms, 
and r the common ratio; then the progrellion is a, ar, ar 2 , 
ar 3 . ar n *, and, lince b is the lad term, ar' 1 i z=zb, 
i 
and r n therefore r—^A ; and, r being known, 
a a\ 
the terms of the progrefiion ar, at 2 , ar 3 , See. areknown. 
To find the fum of a feries of quantities in geometrical 
progrefiion, fubtract the firft term from the produdt of 
the lalt term and common ratio, arid divide the remainder 
by the difference between the common ratio and 'unity. 
Let a be the firft term, r the common ratio, n the number 
of terms, y the 1 aft term, and s the fum of the feries: 
Then a-\-ar -\-ar 2 .... -] -ar n 2 -\-ar n i =zs; 
multiplying both tides by r, 
ar-\-ar 2 -\-at 3 .... -\-ar n 1 -| -ar n —rs 
Sub. a-\-ar-\-ar 2 \-ar 3 .... -j -ar n 1 222s 
Rem. — a-\-ar n z 
ar 1 
Or r=- 
irs- 
—a 
~s—r~ 
ry — 
-i X*. 
Cor. i. From the equation s~— 
ry- 
any three of the 
the vulgar fraction, correfp.onding to fuch a decimal, is 
found by fumming the feries. 
Ex. Required the vulgar fraction correfponding to the 
decimal ‘123123123, &c. 
Let ‘1231231.23, &c. —s; then, multiply both Tides by 
iooo , and 1 23• 123 1 23 1 23, See. =10005 ; and, by fubtraiff- 
ing the former equation from the latter, 1 23=999.?; there - 
r I2 3 4 1 
fore, s— —-=-—. 
999 333 
On Permutations and Combinations. 
The different arrangements that any quantities admit, 
are called their permutations. Thus the permutations of 
a, b c, taken two and two together, are ab, ba, ac, ca, be, cb. 
The combinations of quantities are the different collec¬ 
tions that can be formed out of them, without regarding 
the order in which they are placed. Thus ab, ac, be, are 
the combinations of the quantities, a, b, c, taken two and 
two; ab and ba, though different permutations, forming 
the fame combination. 
The number of permutations which 12 quantities admit, 
taken two and two together,, is 12.12—1; taken three and 
three together, is 12.12— \.n —2. In 12 things, a, b, c, d, Sec.* 
a may be placed before each of the reft, and thus form 
n —1 permutations; in the fame manner, there are 12— 1 
permutations where b ftands firft, and fo of the reft ; there¬ 
fore, there are upon the whole 12.12—1 permutations of 
this kind, ab, ba, ac, ca, See. 
Again, of 12—i things, b , c, d, See. taken two and two 
together, there are 12— i.n —2 permutations, by the for¬ 
mer part of the article, and, by prefixing a to each of 
thefe, there are 22—1.12 —z permutations, taken three and 
three, where a ftands firft; the fame may be faid of b, 
c,d. See. therefore there are upon the whole n.n —1.12—2 
fuch permutations. 
Co r,. By following the fame method, it appears, that in 
12 things, if r of them be-always taken together, there are 
12.1:—1.12—2.12—3.12— r-\-i permutations.. 
The number of combinations that can be formed out of 
and n things, taken two and two together, is n.- — -; or, taken 
three and three together, the number is n.- — 
2 3 _, 
The number of permutations in the firft cafe is n.n —1„ 
but each combination ab, admits of two permutations, 
ab, ba ; therefore there are twice as many permutations as 
combinations, or the number of combinations is n. K 
quantities, s, r,y, a, being given, the fourth may be 
found. 
Cor. 2 . When r is a proper fraftion, as n increafes, 
the value of r n , or of ar n , decreafes, and when n is in- 
creafed without limit, ar n becomes lefs, with refpeft to a, 
than any magnitude that can be aftigned ; and therefore s=z 
•— o- a a 
——-. This quantity, --- is the limit to which 
the fum of the terms approaches, but never aftually at¬ 
tains ; yet, as it differs from the fum by a quantity lefs 
than any that can be afiigned, it may always be fubftituted 
for it without error. 
Ex. To find the fum of 20 terms of the feries, 1, 2, 4, 
%, Sec. ’ 
Here a— 1, r— 2, 1222:20; 
Therefore 5=——- -—2 20 —1. 
2—1 
Recurring decimals are quantities in geometrical pro- 
greffion, where —, -, -, Sec. is the common ra- 
10 100 1000 
tio, according as one, two, three, &c. figures recur; and 
Vol. I. No. 19. 
Again, there are n.n —1.12—2 permutations in h things, 
taken three and three together; and each combination of 
three things admits of 3.2.1 permutations; therefore there 
are 3.2.1 times as many permutations as combinations, and 
confequently the number of combinations is ————- 
Cor. In the fame manner it appears, that the number 
of combinations in 12 things, each of which contains r of 
them, is 
-r -hi 
On the Binomial Theorem. 
The method of railing a binomial to any power, by re ¬ 
peated multiplication, has already been laid down, under- 
Involution and Evolution. A general rule by which this may 
be done much more expeditioully was firft difeovered by 
Sir If. Newton. Let x-\-a be the given binomial; the-12th 
power of it is x n -\-nax n l -J-?2.-— ~a?x n — 8 -{-??. 
cV' 3 -j-, Sec. Where The index of x, beginning from ' 
it, is diminilhed by unity, and the index, of a, beginning 
3 F from 
