sequences

sequences - (12.1) Sequences A sequence is a list of...

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Unformatted text preview: (12.1) Sequences A sequence is a list of numbers. More technically, a sequence is a function with domain the positive integers. We could write f ( n ) for the n-th term (number) of the sequence, but more usually we write something like a n for the n-th term. Sometimes, people write { a 1 ,a 2 ,a 3 ,... } to indicate a sequence, or { a n } n =1 . Other people prefer ( a n ) n =1 . Examples 1. a n = (- 1) n 3 n . So a 1 =- 1 3 , a 2 = 1 9 , a 3 =- 1 27 , and so on. 2. a n = n n 2 + 1 . So a 1 = 1 2 , a 2 = 2 5 , and a 10 = 10 101 . 3. Let p n be the n-th prime number. So p 1 = 2 (the first prime number by custom is 2 not 1), p 2 = 3, p 3 = 5 and p 4 = 7. Euclid proved that there are infinitely many primes, so this makes sense. But no one knows a formula for the n-th prime. 4. Let f 1 = 1, f 2 = 2 and let f n = f n- 1 + f n- 2 for n > 2. So f 3 = f 2 + f 1 = 2 + 1 = 3 and f 4 = f 3 + f 2 = 3 + 2 = 5. These are the Fibonacci numbers. They are defined recursively: the n-th term is defined in terms of the preceeding terms. There is a-th term is defined in terms of the preceeding terms....
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sequences - (12.1) Sequences A sequence is a list of...

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