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progressions

# progressions - Arithmetic and Geometric Sequences Felix...

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Arithmetic and Geometric Sequences Felix Lazebnik This collection of problems 1 is for those who wish to learn about arithmetic and geometric sequences, or to those who wish to improve their understanding of these topics, and practice with related problems. The attempt was made to illustrate some ties between these and other notions of mathematics. The prob- lems are divided (by horizontal lines) into three groups according to di culty: easier, average, and harder. Of course, the division is very subjective. The collection is aimed to freshmen and sophomores college students, to good high school students, and to their teachers. ∗ ∗ ∗ An infinite sequence of numbers { a n } n 1 is called an arithmetic sequence, if there exists an number d , such that for every n 1, a n +1 = a n + d : d n N ( a n +1 = a n + d ) . The number d is called the common di ff erence , or just the di ff erence of the arithmetic sequence. Examples: a 1 = 1 , d = 1 : 1 , 2 , 3 , 4 , 5 , . . . ; a 1 = 12 , d = 8 . 5 : 12 , 20 . 5 , 29 , 37 . 5 , . . . ; a 1 = 5 , d = 1 : 5 , 1 , 3 , 7 , . . . ; a 1 = 2 , d = 0 : 2 , 2 , 2 , 2 , . . . . An infinite sequence of numbers { b n } n 1 is called a geometric sequence, if there exists an number r , such that for every n 1, b n +1 = b n r : r n N ( b n +1 = b n r ) . The number r is called the common ratio , or just the ratio of the geometric sequence. Examples: b 1 = 2 , r = 1 : 2 , 2 , 2 , 2 , . . . ; b 1 = 1 , r = 2 : 1 , 2 , 4 , 8 , . . . ; b 1 = 3 , r = 5 : 3 , 15 , 75 , 375 , . . . ; b 1 = 3 , r = 1 / 2 : 3 , 3 / 2 , 3 / 4 , 3 / 8 , . . . . 1 Last revision: December 20, 2008. 1

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Often an arithmetic (a geometric) sequence is called an arithmetic (a geo- metric) progression . The term comes from Latin progredior – ‘walk forward’; progressio – ‘movement forward’, ‘success’. Problems about progressions go back to Rhind Papyrus , c. 1550 BC and Babylonian astronomical tables, c. 2500-2000 BC. 1. Let { a n } n 1 be an arithmetic sequence with di ff erence d . Prove that (a) a n = a 1 + ( n 1) d, for all n 2 (b) a 1 + a n = a 2 + a n 1 = a 3 + a n 2 = . . . (c) S n = a 1 + a 2 + · · · + a n = a 1 + a n 2 n = 2 a 1 +( n 1) d 2 n for all n 2 (and for n = 1, if we assume S 1 = a 1 ). 2. Let { b n } n 1 be a geometric sequence with ratio r . Prove that (a) b n = b 1 r n 1 , for all n 2 (b) b 1 b n = b 2 b n 1 = b 3 b n 2 = . . . (c) Let n 2, and S n = b 1 + b 2 + · · · + b n . Then S n = n i =1 b i = b 1 b n r 1 r = b 1 (1 r n ) 1 r , if r = 1 nb 1 , if r = 1 . The formula also holds for n = 1, if we assume S 1 = b 1 . (d) If | r | < 1, then lim n →∞ r n = 0, and i =1 b i = b 1 1 r . 3. Given that each sum below is the sum of a part of an arithmetic or geo- metric progression, find (or simplify) each sum. (a) 75 + 71 + 67 + · · · + ( 61). (b) 75 + 15 + 3 + ... + 3 5 7 . (c) 1 + 2 + 3 + . . . + ( n 1) + n . (d) i + ( i + 1) + ( i + 2) + · · · + j , where i, j Z , i < j . (e) 1 + 3 + 5 + · · · + (2 n 1), where n N . (f) x i + x i +1 + x i +2 + · · · + x j , where i, j Z , i < j , x = 1. 4. Suppose that the sum of the first n terms of an arithmetic sequence is given by the formula S n = 4 n 2 3 n for every n 1. Find three first terms of the arithmetic sequence and its di ff erence.
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progressions - Arithmetic and Geometric Sequences Felix...

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