16. The sum of the terms of an A. P. of twelve terms is 354, and the sum of the even terms is to the sum of the odd terms as 32 to 27. What is the common difference? 17. How many positive integers of three digits are there which are divisible by 9? Find their sum. n, and the sum of n terms is -n terms? 18. If the sum of m terms of an A. P. is m, what is the sum of m + n terms? Of m 19. Show that the sum of 2 n + 1 consecutive integers is divisible by 2 n + 1. 20. Prove that if the same number be added to each term of an A. P., the resulting series will be an A. P. 21. Prove that if each term of an A. P. be multiplied by the same number, the resulting series will be an A. P. 22. Prove that if in the equation y : = ax + b, we substitute c, c + d, c + 2 d, ..., in turn for æ, the resulting values of y will form an A. P. 23. Prove that if a2, b2, c2 form an A. P., then 1 1 1 form an A. P. 24. If a, b, c form an A. P., then } (a + b + c)3 = a2 (b + c) + b2 (a + c) + c2 (a + b). 25. A laborer agreed to dig a well on the following conditions: for the first yard he was to receive $2, for the second $2.50, for the third $3, and so on. If he received $42.50 for his work, how deep was the well? 26. On a certain day the temperature rose 1o hourly from 5 to 11 A.M., and the average temperature for that period was 8°. What was the temperature at 8 A.M.? 27. Twenty-five trees are planted in a straight, line at intervals of 5 feet. To water them, the gardener must bring water for each tree separately from a well which is 10 feet from the first tree and in line with the trees. How far has the gardener walked when he has watered all the trees? 28. Two bodies, A and B, start at the same time from two points, C and D, which are 75 feet apart, and move in the same direction. A moves 1 foot the first second, 3 feet the second, 5 feet the third, etc.; B moves 3 feet the first second, 4 feet the second, 5 feet the third, etc. How long will it take A to overtake B ? 29. A number of equal balls are placed in the form of a solid equilateral triangle in the following way: one ball is placed at the vertex, under this are placed two balls, under these two are placed three balls, and so on. If the number of balls is increased by 4, they can be placed in the form of a solid rectangle whose base is equal to the base of the triangle, and whose altitude is 3 balls shorter than the base. How many balls are in the triangle ? § 3. GEOMETRICAL PROGRESSION. 1. A Geometrical Series, or as it is more commonly called a Geometrical Progression (G. P.), is a series in which each term after the first is formed by multiplying the preceding term by a constant number. See § 1, Art. 1, (2). Evidently this definition is equivalent to the statement that the ratio of any term to the preceding is constant. For this reason the constant multiplier of the first definition is called the Ratio of the progression. The five numbers a1, an, r, Sn, n, are called the Elements of the progression. 2. The ratio may be either larger or smaller than 1; in the former case the progression is called a rising or ascending progression; in the latter a falling or descending progression. If the terms are all positive, the words increasing and decreasing may be used for ascending and descending, respectively. 3. In a geometrical progression any term is equal to the first term multiplied by a power of the ratio whose exponent is one less than the number of the required term, i.e., an= By the definition of a geometrical progression. а1 = α1, а2 = α1r, α3 = ɑ«r = α1r12, α1 = α ̧r = α ̧3, etc. (I.) The law expressed by the relations for these first four terms is evidently general, and since the exponent of r is one less than the number of the corresponding term, we have = Ex. 1. If a1 = 1, r = 3, n = 5, then a = · 3* = 31. This relation may also be used to find not only a when a r, and n are given, but also to find the value of any one of the four numbers when the other three are given. Ex. 2. If a1 = 4, a¿ = }, n = 6, then } = 4 r3, whence r = It is important to notice that, while a1, an, and r may be positive or negative, integral or fractional, n must be a positive integer. Consequently a, a,, r cannot be assumed arbitrarily. As yet the value of n can be determined from (I.) only by inspection. Substituting a, for a1-1 in (II.), we have The first forms of (II.) and (III.) are to be used when r<1, 3, r = = 2, n = 6, to find St. 3 (26 — 1) — 189. = Formulæ (II.) and (III.) may be used not only to find S when a1, r, and n, or a, a,, and r are given, but also to find the value of any one of the four elements when the other three are given. 6. Formulæ (I.) and (II.), or (I.) and (III.), may be used simultaneously to determine any two of the five elements, ɑ ɑn, 1, Sn, n, when the three other elements are given. Ex. 1. Given r=— 11, n = 5, 05 From (I.), − 1 = α1( − 1)1, whence a1 =— 4; and from (II.), using the value of a1 just found, = From (III.), 31 = 16 × 2 — α, whence a1 = . From (I.), 2-1 162"-1, whence n = 6. Ex. 3. Given n = 7, a, = 16, S7 = 31, to find r and a1. Thus this example leads to an equation of the seventh degree, which cannot be solved. The value of r in such equations can often be found by inspection. In the above equation r = 2. We then have a1 = 1. 7. In many examples the elements necessary for determining the element or elements directly from (I.)-—(III.) are not given, but in their place equivalent data. Ex. 1. Given a5 = 48, a3 = 384, to find a, and r. 48 art, and 384 = a17; = From (I.), whence 78, or r = 2. Therefore a1 = 3. Or, we could have regarded 48 as the first term and 384 as the last term of a progression of four terms. Then by (I.), Ex. 2. Given S = 63, S, = 511, to find a, and r. We then have from (1): a,1, when r = 2; and a1 = 100 (23/3+3), when r = -2 3/3. Such examples as the last in general lead to equations of higher degree than the second. EXERCISES IV. Find the last term and the sum of the terms of each of the following geometrical progressions : In each of the following geometrical progressions find the values of the elements not given: |