- #1
kathrynag
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Homework Statement
[tex]F_{1}+F_{3}+F_{2n-1}[/tex]=[tex]F_{2n}[/tex]
Homework Equations
The Attempt at a Solution
P(k+1):[tex]F_{2k-1}+F_{2k+1}[/tex]=[tex]F_{2k+2}[/tex]
mutton said:What happened to F_1 and F_3?
kathrynag said:well, it's F_1+F_3+...+F_2n-1
kathrynag said:Homework Statement
[tex]F_{1}+F_{3}+F_{2n-1}[/tex]=[tex]F_{2n}[/tex]
Homework Equations
The Attempt at a Solution
P(k+1):[tex]F_{2k-1}+F_{2k+1}[/tex]=[tex]F_{2k+2}[/tex]
What was "F_1+ F_3+ ...+ F_2n-1"?kathrynag said:well, it's F_1+F_3+...+F_2n-1
kathrynag said:F-1+F-3+...+F_2k-1+F_2k+1
P(k)+F_2k+1
F_2k+F_2k+1
Now I'm stumped...
mutton said:Very close. What happens when 2 consecutive Fibonacci numbers are added?
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding numbers, starting with 0 and 1. The sequence is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.
Proof by induction is a mathematical method used to prove that a statement is true for all natural numbers. It involves two steps: a base case, where the statement is proven true for the first number, and an inductive step, where it is shown that if the statement is true for one number, it is also true for the next number.
Fibonacci proof by induction works by using mathematical induction to prove that the Fibonacci sequence follows a specific pattern. The base case is usually shown by proving that the first two numbers in the sequence are 0 and 1. Then, the inductive step is used to show that if the statement is true for the nth number, it is also true for the (n+1)th number. This proves that the statement is true for all numbers in the Fibonacci sequence.
The key steps in a Fibonacci proof by induction are:
1. Proving the base case, usually by showing that the first two numbers in the sequence are 0 and 1.
2. Assuming the statement is true for the nth number in the sequence.
3. Using the inductive hypothesis to show that the statement is also true for the (n+1)th number in the sequence.
4. Concluding that the statement is true for all numbers in the Fibonacci sequence based on the inductive step.
Fibonacci proof by induction is important because it provides a way to prove that a statement is true for all numbers in the Fibonacci sequence, without having to test every single number. This method is used in many areas of mathematics and computer science to prove statements about patterns or sequences that follow a specific rule.