| XXX | (90) | XXX |
90 ALGEBRA. ‘‘ The fum of a feries of geometrical proportionals qua] quantities you fubtraft the fame quantity, the re“ wanting the firft term, is equal . to the fum of all mainders muft be equal.” “ but the laft term multiplied by the common ratio.” By this, rule, when the known and unknown quantities mixed in an equation, you may feparate them by For ar+ar'--ar&c. + £-+p: +-y+J = are bringing all the unknown to one fide, and the known to the other fide of the equation ; as in the following examl zzrXa-{-ar--ar , &c. fA + ri + ples. r Suppofe 5x+50=4x-f56 Therefore if j be the fnm of the feries, j — a will be e- hy Tranfpofit. jx—4X=56—50, or, x=6 «jual to /—jXr; that is, / — a — sr—yr, ov sr — f — And if 2x+a =x-M , yr—a 2X' — rx or, x~l—a. yr—a, and j = ——-. II. Any quantity by which the unknown Since the exponent of r is always increafing from the Rule is multiplied tnay be taken away, if you difecond term, if the number of terms be n, in the lad quantity vide all the other quantities on both fides of the equaterm its exponent will be n—i. Thereforea
- tion
by it. xT n and —ar ; and/ = f^—?-=z ——So For that is to divide both fides of the equation by the
r—i / r—i
that having the firft term of the feries, the number of fame quantity, and when you divide equal quantities by the terms, and the common ratio, you may eafiiy find the fame quantity, the quotients muft be equal. Thus, the fum of all the terms. If ax=b If it is a decreafing feries whofe fum is to be found, then x—— ; as of A- +A. + ^7> &c. + r?r} + and the number of the terms be fuppofed infinite, then and if 3x+12=2 7 ihall a, the laft term, be equal1 to nothing. For, beby Rule ift. jx—27—12=15 and by Rule 2d. =5 caufe », and confequently r"— is infinite, a=o. Rule III. If the unknown quantity is divided by any The fum of fuch a feries
- which is a finite quantity, that quantity may be taken away if you multiply all the other members of the equation by it.
fum, though the number ,of terms be infinite. Thus, Thus, I +4r + i + -ff-{'-5r5 "K StC. = ^ ~ —2. then fhall x=ii+5i and l+-m-KV + Tr+> By this rule, an equation whereofthat any(hall part beis exprefa frac• t;ion may be reduced to an equation by integers. If there are more fradlions than one in Chap. IX. Of Eq^uat i oks that involve fed the given equation, yon may, by reducing them to a only one unknown Quantity. common denominator, and then multiplying all the other by that denominator, abridge the calculation An equation is “ a propofition afTerting the equality terms “ of two quantities.” It is exprefled moft common- thus; ly by fetting dov/n the quantities, and placing the figa If + = —7 T T * { — ) between them. An equation gives the value of a quantity, when that then 3i±I*=x-7 quantity is alone on one fide of the equation: and that and by this Rule gx-fjxeriyx—105 15 1 value is known, if all thofe that are on the other fide are and by R. 1. and 2. xzz 3.* =15. known. Thus if I find that xzz—— 3 = 8, Iwehavearea toknown IV. If that member of the equation that invalue of x. Thefe are the laft conclufions feek Rule the unknown quantity be a furd root, then the in queftions to be refolved; and if there be only one volves equation is to be reduced to another that fliall be free unknown rquantity in a given equation, and only one di- from furd, by bringing that member firft to ftand menfion o it, fuch a value may always be found by the alone any upon one fide of the equation, and then taking following rules. away the radical fign fropi it, and raifing the other Rule I. Any quantity may be tranfpofed from one fide of the equation to the power denominated by, the fide of the equation to the other, if you change its furd. fign. Thu? if V 4X+16=12 For to take away a quantity from one fide, and to place 4x4-16=144 it with a contrary fign on the other fide, is to fubtratt it and 4X=i44—16=128 from both fides; and it is certain, that “ when from earjd x=,i8=32. Rule
