PROGRESSIOk 
Let a, ar, ar'^ ar\ ar\ &c. be the ijuan- 
tities ; their differences, ar — a, ar^ — ar, 
ar’ — «)■’ — ar* — ar\ &c. form a geo- 
metrical progression, whose first term is 
ar — «, and common ratio n 
“ Quantities in geometrical progression 
are proportional to tiieir differences,” 
For d ; <tr :: ar — a : ar* — ar ar’ — 
ar : «r’ — ar’, &Ci 
“ In any geometrical progression, the 
first term is to ,the thirdj as the square of 
the first to the square of the second.” 
Let a, ar, ai**, &c. be the progression; then 
u : ar* :: a* : a*r*. 
Hence it appears, that the duplicate ratio 
of two quantities (Euc. Def. 10. 5); is the 
ratio of their squares. 
In the same manner it may be shown, 
that the first term is to the n + 1* t*’™* 
the first raised to the n* power, to the se- 
cond raised to the same power. 
“ If any terms be taken at equal intervals 
in a geometrical progression, they will he 
in geometrical progression.” 
Let a, ur-.-ar” ....kur*’ 
. &c. 
Sub. I 
Rem. — «-j-ar” = rs — s — r- 
ar" — a ry — a 
’ r — 1 r — 1 
■1 X s 
From the equation s ; 
VOL. V. 
ry- 
1 1 — r 
w-liicli we call the 
of the quantities, s, r, aj, a, being given, the 
fourth may be found. AVhen r is a proper 
fraction, as n increases, the value of r”, or 
of ar", decreases, and when n is increased 
without limit, ar" becomes less, with re- 
spect to a, than any magnitude that can be 
— ft n 
assigned ; and therefore * = - 
This quantity 
Sum of the series, is the limit to which die 
Slim of the terms approaches, but never ac- 
tually attains ; it is, however, the true re- 
presentative of the series continued sine 
fine ; for this series arises from the division 
a 
of a by 1 — r ; and therefore 
without error, be substituted for it. 
Ex. li To find the sum of 20 terms of 
the series, 1, 2, 4, 8; &c. 
Here a = 1, r = 2, = 20 ; therefore, 
1 y 1 
^ =a2’° — 1. 
2 — 1 
Ex. 2. Required the sum of 12 terms of 
the series 64, 16, 4, &c. 
be the progression, then a, or”, ar*", ar*", 
See. are at the interval of n terms, and form 
a geometrical progression, whose common 
ratio is r". 
“ If the two extremes, and the number 
of terms in a geometrical progression be 
given, the means may be found.” 
Let a and b be the extremes, n the num- 
ber of terms, and r the common ratio ; then 
,.ar"-qand 
Here a — M,r=:-,n: 
4 
: 12, therefore. 
4'’' 
• 64 
64x4” — 64 
_64 4'’— 1 
■411^ ■4—1 ' 
the progression is a, ar, a r, a 
since b is the last term, ar"-' — b, and 
therefore r = - ” * : and r be- 
a « 
ing thus known, the terms of the progres- 
sion ar, ar*, ar*, &c. are known. 
“ To find the sum of a series of quantities 
in geometrical progression, subtract the 
first term from the product of the last term 
and common ratio, and divide the remain- 
der by the difference between the common 
ratio and unity.” 
Let a be the first term, r the common 
ratio, n the number of terms, y the last 
term, and s the sum of the series : 
Then q-j- ar -|-ar’ -j- ar"~’-)- ar"-’ 
= s; and multiply^ both sides by r, 
a r a r’-jjar’. . . .-j-a r’— *-j-a r"= r s 
l>.a-|-ar-i-ar’T|par’....-^ar”— ' =s 
1 
4-^ 
Ex. S. Required the sum of 12 terms of 
the series 1, — 3, 9, — 27, &c. 
In this case, o = 1, r = — 3, n = 12 ; 
therefore, .s = — i 
— o — 1 4 
E.v. 4. To find-the sum of the series 1 — 
2^4 8 
-j- Sec. in infinitum. 
Here a = 1, r = - 
224), ' 
- ; therefore, (.'krt. 
-j, any three 
i 3' 
1 + 2 
Tt may be observed, in connection with 
this subject, that the recurring decimals are 
quantities in geometrical progression, where 
-L, , , Sec. is the common ratio, ac- 
10 100 ’ 1000 ’ ’ 
cording as one, two, three, &c. figures re- 
cur; and the vulgar fraction, correspond- 
ing to such a decimal, is found by summing 
the series. 
Ex. 5. Required the vulgar fraction cor- 
responding to the decimal .123123123, &c. 
Let .123123,123, &c. = s; then, multiply 
both sides by 1000 ; and 123.123123129, 
li 
