7. In any geometrical series, the last term is equal to the product arising from multiplying the first term by such a power of the ratio as is denoted by the number of terms less one. Thus, in the series 2, 6, 18, 54, 162, we shall have 2×34 =2 × 81 = 162. And in the series a, ar, ar”, ar”, ar", &c. continued to n terms, the last term will be l=arm-l. 8. The sum of any series of quantities in geometrical progression, either increasing or decreasing, is found by multiplying the last term by the ratio, and then dividing the difference of this product and the first term by the difference between the ratio and unity. Thus, in the series 2, 4, 8, 16, 32, 64, 128, 256, 512, Or the same rule, without considering the last term, may be expressed thus: Find such a power of the ratio as is denoted by the number of terms of the series; then divide the difference between this power and unity, by the difference between the ratio and unity, and the result, multiplied by the first term, will be the sum of the series. Thus, in the series a+ar-Hara-i-ar3+ars, &c. to armo, we shall have n r 1 S=a r". r ar 1 Where it is to be observed, that if the ratio, or common multiplier, r, in this last series, be a proper fraction, and consequently the series a decreasing one, we shall have, in that case, - a- - g a-i-ar-i-ar” +ar*-ī-ar", &c, ad infinitum = 1 EXAMPLES. 1. The first term of a geometrical series is 1, the ratio 2, and the number of terras 10 ; what is the sum of the series 7 Here 1 X29 = 1 X512=512, the last term. H. 12 × 2- 24 – And to : --wo- = 1023, the sum required. - - . . 1 2. The first term of a geometrical series is 3? the ra 1 tio 3' and the number of terms 5 ; required the sum of 6. A person being asked to dispose of a fine horse, said he would sell him on condition of having a farthing for the first nail in his shoes, a halfpenny for the second, a penny for the third, twopence for the fourth, and so on, doubling the price of every nail, to 32, the nnmber of nails in his four shoes ; what would the horse be sold for at that rate Ans, 4473924l. 5s. 33d. or EQUATIONS. THE DocTRINE of Equations is that branch of algebra, which treats of the methods of determining the values of unknown quantities by means of their relations to others which are known. This is done by making certain algebraic expressions equal to each other (which formula, in that case, is called an equation), and then working by the rules of the art, till • the quantity sought is found equal to some given quantity and consequently becomes known. The terms of an equation are the quantities of which it is composed ; and the parts that stand on the right and left of the sign ==, are called the two members, or sides, of the equation. Thus, if ac-a-Hb, the terms are ar, a, and b ; and the meaning of the expression is, that some quantity ar,” standing on the left hand side of the equation, is equal to the sum of the quantities a and b on the right hand side. A simple equation is that which contains only the first power of the unknown quantity: as, a-Ha-3b, or az=&c, or 2x+3a*=552 ; ---. Where a denotes the unknown quantity, and the other * letters, or numbers, the known quantities. A compound equation is that which contains two or more different powers of the unknown quantity ; as, a 2+ax+b, or a 3–42°-H3x=25. Equations are also divided into different orders, or receive particular names, according to the highest power of the unknown quantity contained in any one of their terms: as, quadratic equations, cubic equations, biquadratic equations, &c. Thus, a quadratic equation is that in which the unknown quantity is of two dimensions, or which rises to the second power ; as, **=20 ; a 2-1-ax=b, or 3x3 +10x=100. A cubic equation is that in which the unknown quantity is of three dimensions, or which rises to the third power : as, a:3 ==27 ; 223–3.c-35; or a 3-ax” +bac=c. A biguadratic equation is that in which the unknown quantity is of four dimensions, or which rises to the fourth power : as, a * =25 ; 5.c4–42=6; or a *-azo-F ba.” – crad. And so on, for equations of the 5th, 6th, and other higher orders, which are all denominated according to the highest power of the unknown quantity contained in any one of their terms. o The root of an equation is such a number, or quantity, as, being substituted for the unknown quantity, will make both sides of the equation vanish, or become equal to each other A simple equation can have only one root ; but every compound equation has as many roots as it contains dimensions, or as is denoted by the index of the highest power of the unknown quantity, in that equation. Thus, in the quadratic equation a 2+2x=15, the root, or value of ac, is either + 3 or - 5 ; and, in the cubic equation 23 – 9x2 +26.x=24, the roots are 2, 3, and 4, as will be found by substituting each of these numbers for 3. '**'. In an equation of an odd number of dimensions, one ^ of its roots will always be real ; whereas in an equation of an even number of dimensions, all its roots may be imaginary ; as roots of this kind always enter into an equation by pairs. - Such are the equations 22-62-1-14=0, and z*-2x” –9x2 + 10x+50=0. (2) (z) To the properties of equations above-mentioned, we may here farther add, 1. That the sum of all the roots of any equation is equal to the coefficient of the second term of that equation, with its sign changed. oR THE RESOLUTION or SIMPLE EQUATIONS, Containing only one unknown Quantity. The resolution of simple, as well as of other equations, is the disengaging the unknown quantity, in all such expressions, from the other quantities with which it is connected, and making it stand alone, on one side of the equation, so as to be equal to such as are known on the other side ; for the performing of which, several axioms and processes are required, the most useful and necessary of which are the following: (a) 2. The sum of the products of every two of the roots, is equal to the coefficient of the third term, without any change in its sign. 3. The sum of the products of every three of the roots, is equal to the coefficient of the fourth term, with its sign changed. 4. And so on, to the last, or absolute term, which is equal to the product of all the roots, with the sign changed or not accord ing as the equation is of an odd or an even number of dimension See, for a more particular account of the general theory equations, Vol. II. of my Treatise on Algebra, 8vo. 1813. (a) The operations required, for the purpose here mentione are chiefly such as are derived from the following simple and e dent principles: " *_1. If the same quantity be added to, or subtracted from, ea of two equal quantities, the results will still be equal ; which the same, in effect, as taking any quantity from one side of equation, and placing it on the other side, with a contrary sign 2. If all the terms of any two equal quantities, be multiplie or divided, by the same quantity, the products, or quotient thence arising, will be equal. 3. If two quantitles, either simple or compound, be equal to each other, any like powers, or roots, of them will also be equal. All of which axioms will be found sufficiently illustrated, by the processes arising out of the several examples annexed to the six different cases given in the text, |