Examples in Geometrical Progression. 260. Corollary. The two equations l = a r n − 1 (r—1) Sa (r" — 1) give the means of determining either two of the quantities a, l, r, n, and S, when the other three are known. But it must be observed, that, since n is an exponent, it can only be determined by the solution of an exponential equation. 261. EXAMPLES. 1. Find the 8th term and the sum of the first 8 terms of the progression 2, 6, 18, &c., of which the ratio is 3. Ans. The 8th term is 4374, the sum is 6560. 2. Find the 12th term and the sum of the first 12 terms of the series 64, 16, 4, 1, †, &c., of which the ratio is Ans. The 12th term is 65536, the sum is 85,655 35 196608 4. Find the sum of the geometrical progression of which the first term is 7, the ratio, and the last term 12. Ans. 121. 5. Find r and S, when a, 7, and n are known. Examples in Geometrical Progression. 6. Find the ratio and sum of the series of which the first term is 160, the last term 38880, and the number of terms 6. Ans. The ratio is 3, the sum is 58240. 7. Find r, when a, l, and S are known. 8. Find the ratio of the series of which the first term is 1620, the last term 20, and the sum 2420. Ans.. 9. Find a and S, when 1, r, and ʼn are known. 10. Find the first term and sum of the series of which the last term is 1, the ratio, and the number of terms 5. Ans. The first term is 16, the sum is 31. 11. Find 7, when a, r, and S are known. 12. Find the last term of the series of which the first term is 5, the ratio, and the sum 62. Ans. S-(S — 1 ) r. 13. Find a, when 1, r, and S are known. Ans. a 14. Find the first term of the series of which the last term is, the ratio, and the sum 6. Ans. 5. Infinite Geometrical Progression. 15. Find a and 7, when r, n, and S are known. 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 r' is greater than unity; and we have Since, then, the number of terms is infinite, the formulas for the last term and the sum become l = arn−1 = a x0 = 0, 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 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 1. Ans. 2. 2. Find the sum of the infinite progression of which the first term is 0.7, and the ratio 0.1. Ans. J. 3. Find r, in an infinite progression, when a and S are known. 4. Find the ratio of an infinite progression, of which the first term is 17, and the sum 18. Ans. 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, 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 n-2 A x2 +В x^-1+C xn−2 + &c. + M = 0. If this equation is divided by A, and the coefficients represented by a, b, &c., m, it is reduced to x2+ax2-1+bxn−2+ &c. + m = 0. 267. Theorem. If any root of the equation is denoted by x', the first member of this equation is a polynomial, divisible by x - x', without regard to the value of x. Proof. Denote x-x' by x1), that is, If this value of x is substituted in the given equation, if P [1] is used to denote all the terms multiplied by [1], or |