Suppose another recursive formula is given by b k =xb k-1 + yb k-2 , where b 0 a

Question

Suppose another recursive formula is given by bk=xbk-1 + ybk-2, where b0 and b1 are given. Determine conditions on x, y for the technique used in questions 1)-6) to work in finding a nice function f such that f(k)=bk for all k

For reference here are 1-6

Find the first 20 Fibonacci Numbers

Find a 2x2 matrix A such that xk=Axk-1 for all k > 0. Thus xk=Akx0

Find matricies P, D such that D is diagonal and A=PDP-1. Then find a formula for Ak.

Use your fomula for Ak to find a formula for xk

Now write down a function f such that f(k)=ak for all k

Check that f(19)=a19

Explanation / Answer

Solutions :

   First 20 Fibonacci Numbers

Fn

Fn

Number F0 0 F1 1 F2 1 F3 2 F4 3 F5 5 F6 8 F7 13 F8 21 F9 34 F10 55 F11 89 F12 144 F13 233 F14 377 F15 610 F16 987 F17 1597 F18 2584 F19 4181   

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