MATH
sequences

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

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(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 formula for them, however: f n = 1 5 1 + 5 2 n - 1 - 5 2 n .

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• Spring '08
• VALDIMARSSON
• Integers, Natural number, Prime number, Fibonacci number

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