44 Slides--Sequences, Induction

44 Slides--Sequences, Induction - CS103A HO# 44...

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CS103A HO# 44 Slides--Sequences, Induction 2/22/08 1 Logic and formal proofs Proving mathematical theorems (number theory) Crypto Sequences and summations Mathematical induction Recursion Combinatorics Functions CS103A Sequences and Summations A sequence is an ordered list, possibly infinite, of elements. We will use the following notation: a 1 , a 2 , a 3 , . . . We also refer to the elements of the sequence as terms , and if a k is a term, then k is its index or subscript . An explicit formula for a sequence shows how to calculate the value of each term from its subscript. For example: 3, 6, 11, 18, 27, . . . a 1 a 2 a 3 a 4 a 5 . . . a k = k 2 + 2 Sequences and Summations A sequence is an ordered list, possibly infinite, of elements. We will use the following notation: a 1 , a 2 , a 3 , . . . We also refer to the elements of the sequence as terms , and if a k is a term, then k is its index or subscript . An explicit formula for a sequence shows how to calculate the value of each term from its subscript. For example: 3, 6, 11, 18, 27, . . . a 1 a 2 a 3 a 4 a 5 . . . a k = k 2 + 2 We can also specify a sequence by stating its starting value and a recursive formula that tells us how to calculate a k from one or more preceding values. a k = a k-1 + 2k – 1 where a 1 = 3 k: 1 2 3 4 5 6 7 8 a k : 7, 11, 15, 19, 23, 27, 31, 35 . .. Explicit formula a k = Recursive formula a 1 = 7 a k = k: 1 2 3 4 5 6 7 8 9 10 a k : 0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682 . .. Explicit formula a k = Recursive formula a 1 = 0 a k = Sum and Product Notation Σ k = 1 n a k means a 1 + a 2 + a 3 + . . . + a n Π k = 1 n a k means a 1 ·a 2 3 · . . . · a n
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CS103A HO# 44 Slides--Sequences, Induction 2/22/08 2 Sum and Product Notation Σ k = 1 n a k means a 1 + a 2 + a 3 + . . . + a n Π k = 1 n a k means a 1 ·a 2 3 · . . . · a n Index of summation Upper limit Lower limit Linearity Property of Sums Σ k = 1 n (c · a k + b k ) = c Σ k = 1 a k + Σ k = 1 b k nn Interesting cases: c = 1 b k = 0 for k = 1,…,n Σ i = 1 Σ j = 1 i j 43 What is the value of ? Σ i = 1 4 Writing out the j sum: (i + 2i + 3i) = Σ i = 1 6i = 6 Σ i = 1 i 44 = 6 · 10 = 60 Factoring out i first: Σ i = 1 Σ j = 1 j 4 3 i = Σ i = 1 6i = 60 4 Arithmetic Progressions An arithmetic progression is a sequence of the form a, a + d, a + 2d, a + 3d, . . . , a + (n – 1) d, . . . Initial term Common difference n th term if we number from 1 Arithmetic Progressions An arithmetic progression is a sequence of the form a, a + d, a + 2d, a + 3d, . . . , a + (n – 1) d, . . . Explicit formula: t k = a + (k – 1) d Recursive formula: t 1 = a t k = t k-1 + d k: 1 2 3 4 5 6 7 8 a k : 7, 11, 15, 19, 23, 27, 31, 35 . .. Explicit formula a k = 7 + (k – 1) 4 = 4k + 3 Recursive formula a k = a k-1 + 4 Arithmetic Progressions
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CS103A HO# 44 Slides--Sequences, Induction 2/22/08 3 Arithmetic Progressions An arithmetic progression is a sequence of the form a, a + d, a + 2d, a + 3d, . . . , a + (n – 1) d, . . . If a = 1 and d = 1, the sequence is 1, 2, 3, 4, . . . Σ k = 1 t k = Σ k = 1 k = n n n (n + 1) 2 The sum of an initial segment of an arithmetic progression is called an arithmetic series .
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44 Slides--Sequences, Induction - CS103A HO# 44...

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