PRO 



PRO 



her of terms, and r the common ratio; 

 then the progression is a, a r 1 , a r3 ........ 



a r ! j and since 6 is the last term, a ; n1 



= 6, and r" 1 = -. therefore r=- 



a' a ' 



and r being thus known, the terms of the 

 progression a r, a r 1 , a r3, &c. are 

 known. 



" To find the sum of a series of quan- 

 tities in geometriai progression, subtract 

 the first term from the product of the last 

 term and common ratio, and divide the re- 

 mainder 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 a4-ar-j-ar*....-f-flrn *-f-a r" * 

 =s ; and multiplying both sides by r, 



--ar3.... 



Sub. a -far 

 Rem. a 

 or T _ ar "~ g r ff~ - fl 



= r 1 x 



From the equation 



r ?/ a 



v"zri an y 



three of the quantities, s, r, y, a, being 

 given, the fourth may be found. When 

 r is a proper fraction, as re increases, the 

 value of rn, or of a m, decreases, and 

 when n is increased without limit, ar* 

 becomes less, with respect to , than any 

 magnitude that can be assigned; and 



therefore s-. 



1 1 _,. 



This quantity j-^;, which we call the 



sum of the series, is the limit to which 

 the sum of the terms approaches, but ne- 

 ver actually attains ; it is, however, the 

 true representative of the series conti- 

 nued sine fine ; for this series arises from 

 the division of a by 1 r ; and therefore 



a 

 I _ r may, without error, be substituted 



for it. 



Ex. 1. To find the sum of 20 terms of 

 the series, 1, 2, 4, 8, &c. 



Here a = 1, r = 2, n = 20 ; therefore, 



Ex. 2. Required the sum of 12, terms 

 of the series 64, 16, 4, &c. 



Here a == 64, r =-, u = 12, there- 



*.- 64x4'' 64 64 



fore, , =; =, _-=- X 



41 ' 



Ex, 3. Required the sum of 12 terms 

 of the series, 1, 3, 9, 27, &c. 

 In this case, a = 1, r = 3, n = 12 ; 



'. A-Ul _ J 311 _ J 



therefore, s = Jii - = - 

 Ex A. To find the sum of the series 1 



Here a = 1, r = ; therefore, (Art. 

 22 4), t 2 



5= - r e= . 



L 3 

 1+2 



It may be observed, in connection with 

 this subject, that the recurring decimals 

 are quantities in geometrical progression, 



where ' ' &c> is the common 



ratio, according as one, two, three, &c, 

 figures recur; and the vulgar fraction, 

 corresponding to such a decimal, is found 

 by summing the series. 



Ex. 5. Required the vulgar fraction 

 corresponding to the decimal .123123123, 

 &c. 



Let .123123,123, &c.=s; then multi- 

 ply both sides by 1000 ; and 123.123123 

 123, &c. = 1000s, and by subtracting the 

 former equation from the latter, 123= 



999,; therefore 5 = 1|=|L. 



PROHIBITION, in law, is a writ pro- 

 perly issuing only out of the Court of 

 King's Bench, being the King's preroga- 

 tive writ ; but, for the furtherance of jus- 

 tice, it may now also be had in some cases 

 out of the Court of Chancery, Common 

 Pleas, or Exchequer, directed to the 

 judge and parties of a suit in an inferior 

 court, commanding them to cease from 

 the prosecution thereof, upon a sugges- 

 tion, that either the cause originally, or 

 some collateral matter arising therein, 

 does not belong to that jurisdiction, but 

 the cognizance of some other court. Up- 

 on the court being satisfied that the mat- 

 ter alleged by the suggestion is suffi- 

 cient, the writ of prohibition immediately 

 issues, 



