514. It is demonstrated in Geometry that All similar solids are to each other as the cubes of their like dimensions. We make a few applications of this principle, referring the pupil to the Section on Mensuration for definitions of pyramids, cones, &c. PROBLEMS. 1. How many times will a cube whose edge is 3 in. contain one whose edge is 1 in. ? Ans. 27 times. 2. If an apple 1 in. in diameter is worth 14, how much is one 2 in. in diameter worth? Ans. 89. How 3. A teamster who had a cart-bed 4 × 6 × 1 ft., made another in which each dimension was increased. does the former compare with the latter? 4. Of two similar pyramids, one has a height of 2 ft. and the other of 5 ft. How do they compare in volume? 5. A Winchester bushel is represented vessel 18 in. in diameter, and 8 in. deep. Ans. 23 53. as a cylindrical What would be the dimensions of a similar vessel that would hold 2 bushels? Ans. Dia., 23.308; depth, 10.08 in. CHAPTER VIII. PROGRESSIONS 515. A Series, or Progression, is a succession of numbers, each of which has the same relation to its immediately preceding number. Such numbers are said to be in progression. 516. The Terms of a Series are the numbers which compose it. 517. An Ascending Series is one composed of terms each of which is larger than the preceding term. 518. A Descending Series is one composed of terms each of which is smaller than the preceding term. 519. The Extremes of a series are the first term and the last term. 520. The Means of a series are all the terms except the first and last. In reference to the ratio of their terms, series are either Arith metical or Geometrical. ARITHMETICAL SERIES. 521. An Arithmetical Series is one whose terms increase, or decrease, by a number called the Common Difference. Thus, 1, 4, 7, 10, &c., is an ascending, and 25, 22, 19, 16, 13, &c., is a descending series, in each of which the common difference is 3. To find the Last Term, when the Common Difference and First Term are given. EXAMPLE.-Find the 10th Term when the 1st term is 2, and the common difference is 3. SOLUTION. 2 + (101) × 3 = 29. EXPLANATION.-It is evident that the second term = the first term once the common differ ence, and the third term = the first term + twice the common difference, and so on. We conclude, therefore, that the tenth term is equal to the first term + nine (10 — 1) times the common difference; and in general that 522. RULE.- Any term equals the first term plus as many times the common difference as there are units in the number of the term less one. If the series is a descending one, we use the word minus for plus in the Rule. Ans. 53. 4. Find the 20th term of 1, 14, 11, 12, &c. 5. If at 6% per annum the amount of $400 is $424, $448, &c., what is the amt. at the end of the 11th year? (No. of terms 12.) Ans. $664. To find the Sum of a Series. EXAMPLE. Find the sum of 5 terms of the series 7, 10, 13, &c. SOLUTION. 7 + 10 + 13 + 16 + 19 = Sum. 19+16 +13 + 10 + 7 = Sum. 26 + 26 + 26 + 26 + 26 = 2 Sums, or 5 x 26; 52065, Sum. er, we have twice the sum of the series = 5 × 26 = 5 x sum of the extremes (or of any two terms equidistant from the extremes), which divided by 2 gives once the sum, 65. We have, therefore, this 523. RULE.—The sum of a series is equal to one-half the sum of the extremes multiplied by the number of terms. PROBLEMS. 1. Find the sum of 10 terms of the series 5, 8, 11, &c. (Find Last Term by 522.) Ans. 185. &c. 2. Find the sum of 16 terms of the series 4, 8, 12, &c. Ans. 544. 3. Find the sum of 12 terms of the series $6, $12, $18, Ans. $468. 4. Find the sum of 100 terms of the series, 4, 1, &c. Ans. 1287. To find the Common Difference, when the Number of Terms and Extremes are known. EXAMPLE.-What is the common difference when the first term is 3, and the 22d term is 45 ? SOLUTION.-Since the last term is equal to the first term plus the common difference multiplied by the number of terms less one, if we subtract 3 from 45, we must have left the common difference multiplied by the number of terms less one. = 453) divided by The number of terms less one is 21; and 42 = 21 gives 2, the common difference. 524. RULE.-Divide the difference of the extremes by the number of terms less one. PROBLEMS. What is the common difference when 1. The first term is 1, and the 21st, 41? 2. The first term is 93, and the 32d, 0 ? 3. The first term is 38, and the 21st, 28? Ans. 2. Ans. 3. |