Generalized Fibonacci Sequence: a n = Aa n-1 + Ba n-2 By: Caroline Chen Advisor: Jacob Matherne

1. Generalized Fibonacci Sequence: an = Aan-1 + Ban-2 By: Caroline Chen Advisor: Jacob Matherne

2. What is the Fibonacci sequence? Recurrence Relation--an equation that recursively defines a sequence: each term of the sequence is defined as a function of the preceding terms Defined as Fn = Fn-1 + Fn-2, with F0 = 0, F1 = 1, and n ≥ 2 Therefore F2 = 1, F3 = 2, F4 = 3 and F5 = 5…

3. What is the closed formula for the Fibonacci sequence? Closed Formula: Fn = , where and and Can be shown: 1 + p = p2 and 1 + q = q2 Via Strong Induction Base Case: When n = 0, the closed formula holds. Substituting n = 1 holds as well. Assume this works for all F0…Fk. Show this works for some Fk+1.

4. Continued… Fk+1 = Fk + Fk-1 = = = = = =

5. What is the generalized Fibonacci sequence? Also a Recurrence Relation Defined as an = Aan-1 + Ban-2, with A and B as coefficients and a0 and a1 as some given seed values i.e. if a0 = 1and a1= 1 and A = 5 and B = 2, then a2 = Aan-1 + Ban-2 = 5(1) + 2(1)=7 a3 = Aan-1 + Ban-2 = 5(7) + 2(1) = 37 What is a4? Answer: a4 = 5(37) + 2(7) = 199

6. What is the closed form for the generalized Fibonacci sequence? Guess that an = rn. Original Equation: an = Aan-1 + Ban-2 Reorganize the Equation: an – Aan-1 – Ban-2 = 0 Use our guess: rn – Arn-1 – Brn-2 = 0 Factor: rn-2(r2 – Ar– B) = 0 Since r ≠ 0, r2 – Ar– B = 0. Using the quadratic formula: r1= r2= an1= an2=

7. Case 1 If A2 + 4B > 0, then there are two real roots: r1 and r2. Because our roots, r1 and r2, are linearly independent, a general formula can be written in the form: Here, the closed formula can be written as:

8. Case 1 continued… Now we want and in terms of the givens (a0, a1, A, and B). By plugging in n = 0 and n = 1, we obtain two equations with two unknowns ( and ). Equation 1: Equation 2: or

9. Case 1 continued… By solving that system of equations, you obtain: and So, the closed formula when A2 + 4B > 0 is:

10. Case 1 continued… Let’s try an example! What is a4 when a0 = 3, a1 = 3, A = 1 and B = 2? Remember: Closed: Recursive: an = Aan-1 + Ban-2 Answer: a4 = 2(2)4 + (1)(-1)4 = 33

11. Case 2 If A2 + 4B = 0, then there is only one distinct root: r = In our case, we can write a general formula in the form: where r is the root. Here, the closed formula can be written as: The general case occurs for a quadratic in the form: an= when there is only one root. The root here is . We use system of equations again like before. Ending with:

12. Case 2 continued… Just like in Case 1, we must put and in terms of givens (a0, a1, A, and B). By plugging in n = 0 and n = 1. We obtain two equations with two unknowns ( and ). Equation 1: or Equation 2:

13. Case 2 continued… By solving that system of equations, you obtain: and So, the closed formula when A2+4B = 0 is:

14. Case 2 continued… Let’s try an example! What is a207 when a0 = 17, a1 = 7, A = 2 and B = -1? Remember: Answer: a207 = (17)(1)207 + (10)(207)(1)207 = 2087

15. Case 3 DO IT YOURSELF!!!!!!!!!!!!!!!!!!

16. summary Closed formula for Generalized Fibonacci Sequence: Case 1 (A2 + 4B > 0): Case 2 (A2 + 4B = 0): Case 3 (A2 + 4B < 0): Figure it out!

17. Thank you.