SECTION XL. PROGRESSION. When a series of numbers is given, each one of which has the same ratio to the number which follows it, the series is called a progression. Progression is Arithmetical or Geometrical. Arithmetical Progression is made by the successive addition or subtraction of a common difference. When the common difference is added to each term in order to make the succeeding one, the series is called an ascending series; as 1, 3, 5, 7, 9, 11, &c. When the common difference is subtracted, the series is called a descending series; as 11, 9, 7, 5, 3, 1. If you know the first term, and the common difference of an Arithmetical Progression, you can write the whole series, for to do this, you have only to add or subtract the common difference for each succeeding term. If the whole series is written out, it is evident you can find by inspection, any particular term, as the 7th, the 15th, the 20th, &c. But, if the series be a long one, this may be a very tedious operation. Suppose the series given above, 1, 3, 5, 7, &c., were continued to 87 terms, and you were required to find what was the last term. By examining the series, you will see that the 2d term the 1st+the common difference; the 3d-1st+twice the common difference; the 4th=1st+3 times the common difference; the 5th 1st+4 times the common difference, &c. Any term whatever equals the 1st term, the common difference, multiplied by a number one less than that which expresses the place of the term. The 87th term in the above series, therefore, is 1+2×86=173. 1. What is the 38th term of the series 1, 3, 5, 7, &c. ? 2. What is the 53d term of the same series? the 91st term? the 89th term? the 107th term? 3. In an arithmetical series, the first term of which is 1, and the common difference 3, what is the 64th term? the 75th term? the 81st term? 4. In the series 1, 5, 9, 13, &c., what is the 40th term? the 67th term? the 80th term? 5. In the series 2, 4, 6, &c., what is the 45th term? What is the 100th term? What is the 200th term? Hence, if you know the number and place of any term, and the common difference, you may find the first or any other term. 6. If the 5th term of an arithmetical series is 13, and the common difference 3, what is the 1st term? What is the 24th term? What is the 191st term? 7. If the 6th term of a series is 77, and the common difference 15, what is the 2d term? What is the 14th term? 8. If the 22d term in a series is 89, and the common difference 4, what is the 10th term? What is the 43d term? By knowing the number and the place of any two terms, we may find the common difference. 9. In a certain series the 4th term is 10, and the 7th term is 19; what is the common difference? 19—10—9; now this difference, 9, is made by the addition of the common difference three times; for 7-4-3; the common difference, therefore, is 9-3-3. 10. In a certain series the 5th term is 9, and the 11th term is 21; what is the common difference? 11. In a certain series the 4th term is 13, and the 9th term is 33; what is the common difference? If we know the 1st term, the common difference, and the number of terms, we can find the sum of all the terms. 12. How many strokes does a clock strike in 24 hours, from noon to noon? We might write down the series, 1, 2, 3, &c., up to 12, which would express the number of strokes in 12 hours, from noon till midnight; we might write the same series again, for the time from midnight till noon; and by adding these numbers together, might obtain the answer. But a much shorter way may be found. To exhibit it we will write the two series thus: 1st series, 1 2 3 4 5 6 7 8 9 10 11 12 from noon till midnight. 2d series, 12 11 10 9 8 7 6 5 4 3 2 1 from midnight till noon. 13 13 13 13 13 13 13 13 13 13 13 13 But 13 is the Sum of both series equal to 12 times 13. sum of the first and last terms; and 12 is the number of terms. Therefore, the sum of the first and last terms, multiplied by the number of terms, gives the sum of all the terms of both series. Half this number will be the sum of one series. 13. What is the sum of the series 1, 4, 7, to 20 terms? First find the 20th term. 14. What is the sum of 50 terms of the series 2, 6, 10? 15. A farmer instructed his boy to carry fencing-posts from a pile to the holes in the ground where they were to be inserted, taking one post at a time; the holes are 12 feet apart, in a straight line, and the pile of posts 30 feet from the first hole; how far must he travel, in carrying to their places 100 posts? 16. If the hours in a whole week were numbered in regular progression, and were struck in this way by the clock, how many strokes must the clock strike for the last hour of the week? What would be the whole number of strokes in the week? If we know the first and last terms and the common difference, we can find the number of terms. 17. The first term of a series is 4; the last term is 19; the common difference is 3; what is the number of terms? The difference between the extremes 19-4, is 15; this, you know, is the common difference 3; taken a certain number of times; 15-3=5; there are then 5 additions of the common difference; now the number of terms is 1 more than the number of times the common difference has been added. To find the number of terms, then Find the difference of the extremes; divide it by the common difference; increase the quotient by 1 for the number of terms. * DESCENT OF FALLING BODIES. 18. A body falling through the air falls, the 1st second, 16.1 feet; in the 2d second, 48.3 feet; in the 3d second, 80.5 feet; how many feet farther does it fall each second than it fell the second before? 19. Taking the answer to the preceding question as the common difference, and 16.1 as the first term of a series, how far will a body fall in 4 seconds? *This is a more exact statement than that made in Part I. See Olmstead's Natural Philosophy. It should be remarked, also, that no allow. ance, in these examples, is made for the resistance of the atmosphere, which always diminishes the speed somewhat, and becomes greater and greater as the speed increases. 20. How far will a body fall in 5 seconds? 21. How far will a body fall in 6 seconds? 22. There is a tower 100 feet high; in how many seconds will a stone fall from the top of it to the ground? Remember, you have the sum and the common difference given; the seconds will be the number of terms. 23. There is a monument 220 feet in height; in how many seconds will a stone fall from the top of it to the ground? 24. If a stone dropped into a well strikes the water in 3 seconds, how far is it to the surface of the water? SECTION XLI. GEOMETRICAL PROGRESSION. A series of numbers such that each is the same part or the same multiple of the number that follows it, is called a geometrical series. The ascending series, 1, 3, 9, 27, is of this kind, for each term is one third of that which succeeds it. So, in the descending series, 64, 16, 4, 1, each term is 4 times the following term. The number obtained by dividing any term by the term before it, is called the ratio of the progression. Thus, in the first of the above examples, the ratio is 3; in the second example it is 4. Let us take the series, 2, 6, 18, 54, and observe by what law it is formed. The ratio is 3; the first term, 2. The second term is 2×3, or the first term X the ratio; the third term is 2×32, or the first term X the second power of the ratio; the fourth term is 2X33, or the first term the third power of the ratio. Thus each term consists of the first term multiplied by the ratio, raised to a power whose index is one less than the number expressing the place of the term. 1. What is the 7th term in the series 1, 4, 16, &c.? 3. A glazier agrees to insert a window of 16 lights for what the last light would come to, allowing 1 cent for the first light, 2 for the second, and so on; what did the window cost? 4. If, in the year 1850, the population of the United States shall be 20000000, and if it shall thenceforward double once in every thirty years, what would be the population in 1970 ? To obtain the sum of the terms, when the first and last terms are given, and the ratio, RULE. Multiply the last term by the ratio, subtract the first term from this product, and divide the remainder by the ratio diminished by one. 5. A gentleman promises his son, 11 years old, one mill when he shall be 12 years old, and, on each succeeding birthday till he is 21 years old, ten times as much as on the preceding birth-day; what will the son's fortune be, without interest, when he is 21 years old? SECTION XLII. MENSURATION OF SURFACES. For the mensuration of the triangle and the parallelogram, when the base and height are known, see Sec. XVII. Part I. To find the area of an equilateral triangle when the sides only are known, Square one side; multiply that product by the decimal .433. To find the circumference of a circle, when the diameter is known, Multiply the diameter by 3.1416. 1. What is the circumference of a circle the diameter of which is 36 feet? 2. What is the circumference of a circular race-course whose diameter is 14 miles? 3. What is the circumference of a wheel the diameter of which is 24 feet, 6 inches? 4. What is the circumference of the earth on the line of the equator, its diameter being 7925.65 miles? To find the diameter of a circle, when the circumference is known, Divide the circumference by 3.1416. |