Machinery's Handbook, 31st Edition
LOGARITHMS 39 The technique of taking the log (either common, natural, or other) of both sides of an equation is used often to solve for unknown exponents, as happens with compounding of interest (see ENGINEERING ECONOMICS on page 143). Arithmetic Sequence An arithmetic sequence (also called an arithmetic progression) is a sequence of numbers in which each term differs from the preceding one by a fixed amount, called the common difference , d . Thus, 1, 3, 5, 7, etc. is an arithmetic sequence where the difference d is 2. Here, the consecutive terms of the sequence are increasing by 2. In the sequence 13, 10, 7, 4, etc., the difference is - 3, and the sequence is decreasing. In any arithmetic progression (or portion of one): a = first term of the sequence, also called the a 1 term l = last term considered, also called a n for the n th term n = number of terms d = common difference S n = sum of n terms The formula for the last term is l = a + ( n - 1) d , or a n = a 1 + ( n - 1) d . The sum of an arith- metic sequence with n terms is S n n ( a + l ) = 2 or n ( a 1 + a n ) 2 . In these formulas, d is positive when the progression is increasing and negative when it is decreasing. When any three of the preceding five quantities are given, the other two can be found by the formulas in the table Arithmetic Sequence Formulas on page 40. Often, however, the desired quantity can be determined by working with the information given. Example 1: In a given arithmetic progression, the first term is 5 and the last term 40, and the difference between terms is 7. To find the sum of the progression, first the number of terms has to be found. This is done by considering the difference between the first and last: 40 - 5 = 35. Dividing this by the difference between terms gives the number of intervals between the terms: 35 ÷ 7 = 5. Finally, adding 1 gives the number of terms in the sequence: n = 5 + 1 = 6. The sum of the sequence is: S n = 2 ( a + l ) = (5 + 40) = 3(45) = 135 A geometric sequence or progression is a sequence of numbers in which each term is derived by multiplying the preceding term by a constant multiplier called the ratio . When this ratio is greater than 1, the progression is increasing; when less than 1, it is decreasing. Thus, the sequence 2, 6, 18, 54, etc. is an increasing geometric sequence with a ratio of 3, and the sequence 24, 12, 6, etc. is a decreasing sequence with a ratio of 1 ⁄ 2. In any geometric progression (or part of progression): a = first term of the sequence l = last (or n th) term of the sequence n = number of terms r = ratio of the progression S n = sum of n terms The general formulas for the n th term: l ar n – 1 and = S rl a – r – 1 = ------- When any three of the preceding five quantities are given, the other two can be found by the formulas in the table Geometric Sequence Formulas on page 41. Geometric progressions are used for finding the successive speeds in machine tool drives, and in interest calculations. Example 2 : The lowest speed of a lathe is 20 rpm. The highest speed is 225 rpm. There are 18 speeds. Find the ratio between successive speeds. Ratio r l a -- n – 1 225 20 ----- 17 11.25 17 1.153 = = = = 6 2 Geometric Sequence
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