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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
(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 .
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern