A Practical System of Algebra, by P. Nicholson and J. Rowbotham (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1824 Excerpt: ...series 3069, the ratio 2, and the first term 3. Required the last term? rt--a =--1 s(r--1) = rt--a, .v s(r-l)+a 3069 + 3, s(r-l)+a=rt, or t=-----=----= 1536. (3.) Given the last term 1536, the ratio 2, and the sum of the series 3069. Required the first term? (1.) Given the ratio 2, the first term 5, and the last term 640. Required the sum of the series I Ans. 1275. (2.) Given the sum of the series 1275, the ratio 2, and the first term 5. Required the last term. Ans. 640. (3.) Given the last term 640, the ratio 2, and the sum of the series 1275.' Required the first term. Ans. 5. (4.) Given the sum of the series 1275, the first term 5, and the last term 640. Required the ratio. Ans. 2. PROPOSITION V.--THEOREM. All the products under two corresponding means are equal, and if the progression be odd the square of the middle term is equal to any one of the products. For in the progression, '" a, ar, ar2, ar', ar4, arb, at." The product of the second and sixth terms is ar x ar = or3 X ar3 =5 -V. 6n tfjr Transformation of (Equations. TO transform an equation into another whose roots may be any multiple of the roots in the given equation. Let the roots of the equation x3--px2 + qx--r = 0, be 2, A, c, it is required to transform it into another whose roots shall be ma, mb, mc. Assume v = rax, then will x =--. m v Substitute--for x in the proposed equation, and there will arise Take away the denominators by multiplying all the terms by m, and it becomes v3--pmu2 + qmPv--rm3-0. Where "we may observe, generally, that when an equation is complete we have only to multiply the terms, beginning at the second, by the consecutive powers of the coefficient of the first term, or any assumed multiplier; and this, it is obvious, will extend to equ...