20. When a train arrives at the top of a long slope, the last car is detached and begins to descend, passing over 3 feet in the first second, three times 3 feet in the second second, five times 3 feet in the third second, etc. At the end of 30 seconds it reaches the bottom of the slope. Find its velocity the last second. 21. Insert eleven arithmetical means between 1 and 12. 22. The first term of an arithmetical series is 3, and the sum of six terms is 28. What term will be 9? 23. The arithmetical mean between two numbers is 10, and the mean between the double of the first and the triple of the second is 27. Find the numbers. 24. The first term of an arithmetical progression is 3, and the third term is 11. Find the sum of seven terms. 25. A common clock strikes the hours from 1 to 12. How many times does it strike every 24 hours? 26. The Greenwich clock strikes the hours from 1 to 24. How many times does it strike in 24 hours? 27. Find three numbers in arithmetical progression of which the sum is 21, and the sum of the first and second of the sum of the second and third. Let x y, x, and x + y stand for the numbers. 28. The sum of three numbers in arithmetical progression is 33, and the sum of their squares is 461. Find the numbers. 29. The sum of four numbers in arithmetical progression is 12, and the sum of their squares is 116. What are the numbers? Let x-3y, xy, x+y, and x + 3y stand for the numbers. Geometrical Progression. 362. A series is called a geometrical series or a geometrical progression when each succeeding term is obtained by multiplying the preceding term by a constant multiplier. The general representative of such a series is in which a is the first term and r the constant multiplier, or ratio. The terms increase or decrease in numerical magnitude according as r is numerically greater than or numerically less than unity. 363. The nth Term. Since the exponent of r increases by one for each succeeding term after the first, the exponent is always one less than the number of the term, so that the nth term is arn-1. If the nth term is represented by l, we have 364. Sum of the Series. If represents the nth term, a the first term, n the number of terms, r the common ratio, and s the sum of n terms, then, 365. When r<1, formula II may be more conveniently By putting rl for ar" in formula II, we have (§ 363) When r<1, formula III is more conveniently written 367. From the two formulas I and II, or the two formulas I and III, any two of the five numbers a, r, l, n, s may be found when the other three are given. 1. Find the tenth term of a geometrical series if the first term is 3 and the ratio 2. Here a = 3, r = 2, n = 10. 1 = 3 × 29 = 3 × 512 = 1536. 2. Find the geometrical series if the third term is 20 and the sixth term 160. a = the first term, and r = the ratio. ar2 = the third term, and ar5 = the sixth term. Let Then, 3. Find the sum of six terms of the series 3, 6, 12 ... 4. The first term of a geometrical series is 3, the last term 192, and the sum of the series 381. Find the number of terms and the ratio. 368. The geometrical mean between two numbers is the number which stands between them, and makes with them a geometrical series. If a and b denote two numbers, and G their geometrical mean, then a, G, b are in geometrical progression, and by the definition of a geometrical series (§ 362), 369. Sometimes it is required to insert several geometrical means between two numbers. Insert three geometrical means between 3 and 48. Here the whole number of terms is five; 3 is the first term and 48 the fifth. The terms in brackets are the means required In working out the following results, the student should make use of formulas I, II, and III. |