1 / 17

Expressing Sequences Explicitly

Expressing Sequences Explicitly. By: Matt Connor Fall 2013. Pure Math Analysis Calculus and Real Analysis Sequences. Sequence- A list of numbers or objects in a specific order 1,3,5,7,9,..... Finite Sequence- contains a finite number of terms 2,4,6,8

macon
Download Presentation

Expressing Sequences Explicitly

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Expressing Sequences Explicitly • By: Matt Connor • Fall 2013

  2. Pure Math • Analysis • Calculus and Real Analysis • Sequences

  3. Sequence- A list of numbers or objects in a specific order • 1,3,5,7,9,..... • Finite Sequence- contains a finite number of terms • 2,4,6,8 • Infinite Sequence- contains an infinite number of terms • 2,4,8,16, ........

  4. Arithmetic Sequence- add or subtract a constant to get from one term to the next • 88, 77, 66, 55,....... • Geometric Sequence- multiply or divide by a common ratio to get from one term to the next • 6, 12, 24, 48,........

  5. Recursive Formula- formula for a sequence that relates the previous term(s) to find the new one. • ex: An = A(n-1)+ 4 • Explicit Formula- formula that finds any term in the sequence without knowing any other terms. • ex: An = 1+ 2(n-1) • all you need to know is n

  6. Arithmetic Sequences • General Forms • Recursive formula • An = A(n-1) + d • Explicit formula • An = A1 + d(n-1)

  7. Geometric Sequences • General Forms • Recursive formula • An = r(An-1) • Explicit formula • An = A1 (rn-1)

  8. What about sequences that are not arithmetic or geometric? • This means they do not have a common constant or ratio • These are commonly called Fibonacci-type • The difficult thing about these is finding an explicit formula

  9. Fibonacci Sequence Explicit Formula • Now we will go through deriving an explicit formula for the Fibonacci Sequence • We know the relational formula is • An = An−1 + An−2 • We guess an explicit formula of the form An =Cxn and plug it in to the relational equation and get • Cxn= Cxn−1 + Cxn−2

  10. Cxn = Cxn−1 + Cxn−2 this will always simplify to an equation with the same coefficients as the relational equation, • x2 = x + 1 • Then we collect the terms on one side to use the quadratic formula. • x2 −x−1=0

  11. The quadratic formula gives us x=(1/2)(1±√5) • Therefore: An= B((1/2)(1+√5))n + C((1/2)(1-√5))n • Next we use the first two Fibonacci numbers to find two equations representing B and C • A0=1 and A1=1

  12. This gives us two equations for B and C • B+C=1 and • B(1/2)(1+√5) + C(1/2)(1-√5)=1 • Then we simplify the second equation we have • (B + C) + (B - C)√5 = 2 and since our first equation tells us that B+C=1 we can replace that.

  13. 1 + (B-C)√5 = 2 • We then further simplify this to get the second of our two equations • B+C=1 and B-C=1/√5 • If we add these two equations and simplify we can then solve for B • B= (√5+1)/(2√5)

  14. And then insert the value of B to find the value of C • C=(√5-1)/(2√5) • One More Step!!

  15. If we replace the B and C in our equation for An • This is Binet’s formula, an explicit formula for finding the nth Fibonacci number. An=

  16. As you have seen finding an explicit formula for the nth term in a Fibonacci-type sequence is much more difficult. • . . . . . but they are possible to find!

  17. Resources • http://www.kenston.k12.oh.us/khs/academics/math/AA_11-3A_geometric_sequences_explicit.pdf • http://www.kenston.k12.oh.us/khs/academics/math/AA_11-2A_arithmetic_sequences.pdf • http://www.geom.uiuc.edu/~demo5337/s97b/fibonacci.html • http://faculty.mansfield.edu/hiseri/MA1115/1115L30.pdf

More Related