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 ALGEBRA. 89 4< then thofe quantities are faid to be in geometrical terms.” Thus, a, a--i, e, e--b, are arithmetical ““ proportion.' " Such are the numbers 2, 6, 4, 12; proportionals, and the fum of the extremes is and the quantities equal to; the fum of the mean terms {a-{-b--e). Hence, ter this manner; a, ar, b, br ; which are expreifed afto find the fourth quantity arithmetically proportional to any three given quantities; “ Add the fecond and third, 2 : 6 :: 4 : 12. “ and from their fum fubtradt the firlt term, the remaina : ar :: b : br. “ der (hall give the fourth arithmetical proportional re- And you read them by faying, As 2 is to 6, fo is 4 to 12 j “ quired. ” a is to ar, fo is b to br. In a feries of arithmetical proportionals, “ the fum of or, asfour quantities geometrically proportional, “ the “ the firft and laft terms is equal to the fum of any two “ Inproduct of the extremes is equal to the produdl of the “ terms equally diflrant from the extremes.” If the firft “ middle terms.” Thus, axbr=arxb. And, if it is terms are a, a+b, a-{-2b, <bc. and the laft term x, the required a fourth proportional to any three given laft term but one will be ,v—6,^the laft but two x—2b, quantities,to “findmultiply by the third, and dithe laft but three x—%b, So that the firft half of “ vide their produd bythethe fecond firft, the quotient (ball give the terms, having thofe that are equally diftant from the “ the fourth proportional required.” Thus, to find a laft term fet under them, will (land thus; fourth proportional to a, ar, and b, multiply ar by b, a, a+b, a+2b, a+^b, a+qb, „■ and divide the produd arb by the firft term a, the quox, x—b, x—2b, x—3b, x—4b, tient br is the fourth proportional required. In calculations it fometimes requires a little care to a+x,a-4-x, a-j-x, a + x, a + x, &c. place the terms in due order; for which.you may obAnd it is plain, that if each term be added to the term ferve the following above it, the fum will be a-j-x, equal to the fum of the 'Rule. Firft fet down die quantity that is of the fame firft term a and the laft term. x. From which it is plain, with the quantity fought; then confider, from the that “ the fum of all the terms of an arithmetical pro- kind nature of the queftion, whether that which is given is “ greflion is equal to the fum of the firft and laft taken greater or lefs than that which is fought; if it is “ half as often as there are termsthat is, the fum'of greater, place the greateft of the other two quanan arithmetical progreflion is equal to the fum of the firft tities on then the left hand; but if it is lefs, place the lead and laft terms multiplied by half the number of terms. of the other quantities on the left hand, and Thus, in the preceding feries, if « be the number of the other on thetworight. terms, the fum of all the terms will be « + x X —2 ‘ Then (hail the terms be in due order; and you are to The common difference of the terms being b, and b proceed according to the rule, multiplying the fecond by not being found in the firft term, it is plain that “ its the .third, and dividing their produd by the firft. “ coefficient in any term will be equal to the number of Examp. “ If 30 men do any piece of work in 12 • ‘ terms that precede that term ” Therefore in the laft days, how many men (hall do it in 18 days?” term x youmuft have 71—1 X£, fo that x muft be equal “ Becaufe it is a number of men that is fought, firft fet to a--n— 1 X£. And the fum of all the terms being down 30, the number of men that is given: you will eafily fee that number that is given is greater than the a -fr-xX—•, it •will alfo be equal 10 —or number that isthefought; therefore place 18. on the left hand, and 12 on the right; and find a 4th proportional to a- ^—X«. Thus for example, the feries 1+2 to 18,„ 30, 12, viz.. —--3 30 X — 12 20, +3+4+5, &c. continued to a hundred, muft: be equal When a feries of quantities—increafe by one common 2 XIOO+ IOOOO IOO to —=5050. multiplicator, or decreafe’ by one common divifor, they in geometrical 1 3 proportion Tf a feries have {o) nothing for its firft term, then are faidAs,to a,be ar, ar, ar , ar*, ar , See. or, “ its fum (hall be equal to half the produdt of the laft “ term multiplied by the number of terms.” For then a — — — a a Sc a being = 0, the fum of the terms, which is in general common multiplier or divifor is called their comrt + xx —, will in this cafe be—. From which it monTheratio. is evident, that “ the fum of any number of arithmetical ce In fuch a feries, “ the produd of the firft and laft is always equal to the produd of the fecond and laft £<“ proportionals beginning from nothing, is equal to half the fum of as many terms equal to the greateft term.” “ but on.e, or to the produd of any two terms equal“ 2ly remote from the extremes.” In the feries, a, ar, Thus, ar, ar3, &c. if y be the laft term, then (hall the four o+i+2+3+4+5+6+7+8+9= _ 9+9+9+9+9+9+9+9+9+9 _ioX9__ laft terms of the feries be r, A. A1 A3 7• now it is ~ .2 2 r r 7 “ If of four quantities the quotien t of the firjl and fe- plain, that aXy—arY,, &c. ' cond be equal to the quotient of the third and fourth, Vol. I No. 4. 3 Z “ The
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