What is a proof that every positive integer can be written as the sum of one or more distinct non consecutive Fibonacci numbers?
Prove Zeckendorf’s Theorem: Every positive integer can be written in a unique way as the sum of non-consecutive Fibonacci numbers. (For example, 86 = 55 + 21 + 8 + 2. This is unique, because 55 + 21 + 5 + 3 + 2 has consecutive Fibonacci numbers 2 and 3 (and also 3 and 5).)
What do you notice about every fourth Fibonacci number?
Every 4th number is a multiple of 3 (3, 21, 144.)
What is the Fibonacci of 4?
Fibonacci Numbers List
| F0 F 0 = 0 | F10 F 10 = 55 |
|---|---|
| F4 F 4 = 3 | F14 F 14 = 377 |
| F5 F 5 = 5 | F15 F 15 = 610 |
| F6 F 6 = 8 | F16 F 16 = 987 |
| F7 F 7 = 13 | F17 F 17 = 1597 |
How is the Fibonacci sequence related to the golden ratio?
Two numbers are in the golden ratio if the ratio of the sum of the numbers (a+b) divided by the larger number (a) is equal to the ratio of the larger number divided by the smaller number (a/b). In fact, the higher the Fibonacci numbers, the closer their relationship is to 1.618.
How to prove that a(n) holds for all positive integers n?
Let A(n) be an assertion concerning the integer n. If we want to show that A(n) holds for all positive integer n, we can proceed as follows: Induction basis: Show that the assertion A(1) holds. Induction step: For all positive integers n, show that A(n) implies A(n+1). 3 Standard Example
How do you prove a factorial if n = 5?
Yes, if n = 5 then you want to prove there is a run somewhere of at least 5 consecutive composite integers (though if the run is longer that’s fine). Review the definition of factorial: n! = 1 × 2 × 3 × … × ( n − 1) × n.
How do you prove that b(n+1) holds?
Expanding the right hand side yields n3/3 + 3n2/2 + 13n/6 + 1 One easily verifies that this is equal to (n+1)(n+2)(2(n+1)+1)/6 Thus, B(n+1) holds. Therefore, the proof follows by induction on n. 8 Tip How can you verify whether your algebra is correct?
When to use the inductive hypothesis in a proof?
Fallacy: In the proof we used the inductive hypothesis to conclude max {a − 1, b − 1} = n 㱺 a − 1 = b − 1. However, we can only use the inductive hypothesis if a − 1 and b − 1 are positive integers.