PROGRESSIONS. 448. A progression is a series of numbers increasing or decreasing according to some fixed law. 449. An arithmetical progression is a series of numbers which increase or decrease by a common or constant difference. Thus, 5, 7. 9, 11, 13, 15, is an ascending series, 15, 13, 11, 9, 7, 5, is a descending series, in each of which 2 is the common difference. 450. A geometrical progression is a series of numbers, which increase or decrease by a common or constant ratio. Thus, 1. 3, 9, 27, 81, 243, is an ascending series, 243, 81, 27, 9, 3, 1, is a descending series. In each series the ratio is 3. 451. The terms of a progression are the numbers o which it consists. The first and last terms are called the extremes, and the other terms the means. 452. The following are the elements considered in arithmetical progression and the symbols used: 1. The first term, (a). 3. The common difference, (d). 2. The last term, (). 4. The number of terms, (n). 5. The sum of all the terms, (s). 453. Any three of these quantities being given, the other two may be found. 1) × d. a = l + (n 1) x d. The first term is equal to the last term minus or plus the product of the number of terms less 1, by the common difference, according as the series is ascending or descending. 1. n Ascending series. 2. 1 = 3. d = { The last term is equal to the first term, plus or minus the product of the common difference by 1 less than the number of terms according as the series is ascending or descending. 4. n = a+dx (n-1). Ascending series. a 1 or, d = a + l 5. 8 = 2 a The common difference is equal tremes divided by the number of terms less 1. n xn. + 1; or, n = 1 to the difference of the ex d The number of terms is equal to the difference of the extremes divided by the common difference, plus 1. EXERCISES. The sum of the terms of an arithmetical series is equal to one half the sum of the extremes, multiplied by the number of terms. WRITTEN 454. 1. The first term of an increasing progression is 8, the common difference 5, and the number of terms 20. What is the last term? 2. The first term of a decreasing progression is 203, the common difference 5, and the number of terms 40. What is the last term? 3. What is the 13th term of a descending series whose first term is 75, and common difference 5? 4. What is the amount of $100, at 7%, for 45 years? 5. The first term is 2, the last term is 17, and the number of terms is 6. What is the common difference? 6. The amount of $800 for 60 years, at simple interest, 18 $4160. What is the rate per cent ? 7. The extremes are 7 and 43, and the common difference is 4. What is the number of terms? 8. The first term is 21, the last term is 40, and the common difference is 7. What is the number of terms? 9. In what time will $500, at 7% int., amount to $885? 10. The extremes are 5 and 32, and the number of terms 12. What is the sum of all the terms? 11. How many strokes does a common clock make in 12 hours? 12. What debt can be discharged in a year by weekly payments in arithmetical progression, the first being $24, and the last $1224? 455. The following are the elements considered in geometrical progression: 1. The first term (a). 2. The last term (7). 456. Any three of these quantities being given, the other two may be found. The following formulas or rules cover all the cases: ι 3. The ratio (r). 4. The number of terms (n). 5. The sum of all the terms (s). (7÷pm-1. Ascending series. In an ascending series divide, and in a descending series multiply the last term by that power of the ratio whose exponent is 1 less than the number of terms. 2. l = a × pn-1; or l = a ÷ pn−1 ̧ In an ascending series multiply, and in a descending series divide the first term by that power of the ratio whose exponent is 1 less than the number of terms. The ratio is equal to that root of the quotient of the last term divided by the first, whose index is 1 less than the number of terms. 4. pn-1 = ι a The number of terms less 1 is equal to the exponent of the power to which the ratio must be raised to be equal to the quotient of the last term divided by the first. 1 x r 5. 8 = 2' The sum of a geometrical series is equal to the difference between the product of the last term by the ratio and the first term, divided by the ratio less 1. 1 a WRITTEN EXERCISES. 457. 1. The last term is 192, the ratio 2, and the number of terms 7. What is the first term? 2. The first term of a geometrical series is 6, the ratio 4, and the number of terms 6. What is the last term. 3. The extremes are 2 and 512, and the number of terms is 5. What is the ratio? 4. The extremes are 2 and 1458, and the ratio is 3. is the number of terms? What 5. The extremes are 3 and 384, and the ratio is 2. What is the sum of the series? 6. If a man were to buy 12 horses, paying 2 cents for the first horse, 6 cents for the second, and so on, what would they cost him? 7. A drover bought 7 oxen, agreeing to pay $3 for the first ox, $9 for the second, $27 for the third, and so on. What did the last ox cost him? MENSURATION. 458. Mensuration treats of the measurement of lines, surfaces, and solids or volumes. PLANE FIGURES.* 459. A polygon is a plane figure bounded by straight lines. The area of a plane figure is the surface included within the lines which bound it. A polygon takes its name from the number of sides which bound it, as follows: ооо Heptagon. Trigon. Tetragon. Pentagon. Hexagon. Octagon. The perimeter of a polygon is the sum of its sides; the base is the side on which it stands; and the altitude is the perpendicular distance between its base and a side or angle opposite. 460. A trigon or triangle is a polygon bounded by three sides, having three angles. The base of a triangle is the side on which it is supposed to stand, as AB; the vertical angle is the angle opposite the base, as C; and the altitude is the perpendicular line drawn from the vertical angle to the base, as CD. A Ꭰ 461. A tetragon or quadrilateral is a polygon bounded by four straight lines. B Parallel lines are lines which are equally distant from each other at every point. *Lines, angles, and plane figures, in part, are defined on pages 158, 159. |