Infinite Geometrical Progression. 15. Find a and I, when r, n, and S are known. (r - 1) S Ans. a = gone -1 (7" — 79–1) S 1 = gone -1 16. Find the first and last terms of the series of which the ratio is 2, the number of terms 12, and the sum 4095. Ans. The first term is 1, the last term 2048. 262. An infinite decreasing geometrical progression is one in which the ratio is less than unity, and the number of terms infinite. 263. Problem. To find the last term and the sum of the terms of an infinite decreasing geometrical progression, of which the first term and the ratio are known. Solution. Since r is less than unity, we may denote it by a fraction, of which the numerator is 1, and the denominator p is greater than unity; and we have 1 1 g100 Since, then, the number of terms is infinite, the formulas for the last term and the sum become = a 7-1 = a X0 = 0, r1— а a Examples in Geometrical Progression. that is, the last term is zero, and the sum is found by dividing the first term by the difference between unity and the ratio. 264. Corollary. From the equation a r s 1 either of the quantities a, r, and s may be found, when the other two are known. 265. EXAMPLES. 1. Find the sum of the infinite progression, of which the first term is 1, and the ratio d. Ans. 2. 2. Find the sum of the infinite progression of which the first term is 0.7, and the ratio 0:1. Ans. 7. 3. Find r, in an infinite progression, when a and S are known. Ans. "=1-5 a 4. Find the ratio of an infinite progression, of which the first term is 17, and the sum 18. Ans. It 5. Find a, in an infinite progression, when r and S are known. Ans. a = S(1-r). 6. Find the first term of an infinite progression, of which the ratio is s, and the sum 6. Ans. 2. Form of any Equation. CHAPTER VIII. GENERAL THEORY OF EQUATIONS. SECTION I. Composition of Equations, 266. Any equation of the nth degree, with one unknown quantity, when reduced as in art. 118, may be represented by the form A 2n + B"-1 +C **-%+ &c. + M=0. If this equation is divided by A, and the coefficients B C M Α' Α' A represented by a, b, &c., m, it is reduced to 3* + ax"-1 +6xn-2 + &c. + m=0. &c., 267. Theorem. If any root of the equation an tåxn-1 + b2n–2+ &c.+m= 0 is denoted by x', the first member of this equation is à polynomial, divisible by x - x', without regard to the value of x. Proof. Denote 2 - X by x[1], that is, q[l] = x 2, or I= + 211) If this value of x is substituted in the given equation, if P [1] is used to denote all the terms multiplied by x[1], or Form of any Equation. by any power of xl1), and Q used to denote the remaining terms, the equation becomes P 211 +Q=0. Now the given equation is, by hypothesis, satisfied by the value of x. x = x', or = 0. 2 [1] = 0; by which the preceding equation is reduced to Q The terms not multiplied by x[1], or a power of x [l), must, therefore, cancel each other; and the first member of the given equation becomes P [1], which is divisible by all), or its equal t – 268. Corollary. The preceding division is easily effected by subtracting from the polynomial 2n + am-1+62-2+ &c. + m, the expression 2'n + azn-1 + b win–2 +&c. + m=0, which does not affect its value, but brings it to the form m—x'n + a(zn-1—2n-1) + 6(x*— 2 — x'n-2)+&c., of which each term is, by art. 49, divisible by 2 - x'. The quotient is, by art. 50, 21-1 + x' \n-2 X' 2 n-3 + 23121–4 + &c. ta a x' tax'a 6 bx' to 269. Corollary. The first term of the preceding quotient is xn-1; and if the coefficients of un-2, x*—3, &c., are denoted by a', b', &c., the quotient is xn-1 ta'x”—2 +- b'xn-3+ &c.; α Number of the Roots of an Equation. and the equation of art. 267, is (2 - x')(xn-1 ta' in -2 +6r-3+ &c.)=0; which is satisfied either by the value of x, X = X', or by the roots of the equation mn-1 ta' xn— 2+b'x-3+ &c. = 0. If now" is one of the roots of this last equation, we have in the same way 29–1 + a'2n–2+&c.=(x– ')(x2–2 + a" 2n–3+ &c.)=0, and the given equation becomes (x - x') (1 - 3") (2:"~2+ a2-3+ &c.) = 0; which is satisfied by the value of x'', = x"; so that x" is a root of the given equation. By proceeding in this way to find the roots d'", x", &c., the given equation may be reduced to the form (x — 2') (x — 3") (x — 3") (2 — 2"V)&c=0, in which the number of factors x d', &c. is the same with the degree n of the given equation; and, therefore, the number of roots of an equation is denoted by the degree of the equation ; that is, an equation of the third degree has three roots, one of the fourth degree has four roots, &c. But all these roots are not necessarily real or rational; they may, on the contrary, be irrational or even imaginary. 270. Scholium. Some of the roots x', x'', x'", &c., are often equal to each other, and in this case the number of unequal roots is less than the degree of the equation. |