I understand how to use a calculator to write the decimal point using the golden ratio which is 1.618034 and the conjugate is -.618034. I also know that the relationship between the decimal expansions is that everything to the right of the decimal is the same except one is neg. and one is pos.
Now I have a problem. I have to use the Binet form to calculate the 20th Fib. number. I can use my decimals that I used in the problem above, but I am not sure how to do this. I have worked out the Fibonacci numbers and found that the 20th number is 6,765.
I could really use some help in understanding my problem.
You have correctly calculated the 20th fibonacci number as 6765. That's an excellent start.
To calculate any Fibonacci number without having to go through the recursive process (n times) is to use the Binet's formula.
To simiply things, two constants have been defined:
φ=(1+sqrt(5))/2, and
ψ=(1-sqrt(5))/2
then the nth Fibonacci number can be found with the following formula:
F(n)=(φ^n-ψ^n)/sqrt(5)
If your calculator does not have constants to store φ and ψ you can calculate directly as:
F(n)=(((1+√5)/2)^n-((1-√5)/2)^n)/√5
It should give 6765 if you put n=20.
To calculate the 20th Fibonacci number using the Binet formula, you will need to use the golden ratio and its conjugate. The Binet formula states that the nth Fibonacci number (F_n) can be found using the equation:
F_n = (phi^n - psi^n) / sqrt(5)
where:
- phi is the golden ratio (1.618034)
- psi is the conjugate of the golden ratio (-0.618034)
- n is the desired Fibonacci number
Let's calculate the 20th Fibonacci number step-by-step:
1. Calculate phi raised to the power of 20:
phi^20 = 1.618034^20
To do this calculation, you can use a calculator or a programming language capable of handling floating-point calculations. Enter "1.618034" into your calculator, raise it to the power of 20, and record the result.
2. Calculate psi raised to the power of 20:
psi^20 = (-0.618034)^20
Again, use your calculator to perform this calculation and record the result.
3. Subtract psi^20 from phi^20:
phi^20 - psi^20
Subtract the value calculated in step 2 from the value calculated in step 1.
4. Divide the result by the square root of 5:
(phi^20 - psi^20) / sqrt(5)
Perform this final calculation to get the 20th Fibonacci number. The result should be 6765, which matches your previous finding.
Remember, to use the Binet formula, you need to use the actual decimal values for phi and psi (1.618034 and -0.618034, respectively) instead of rounded values for maximum accuracy.
If you encounter any issues with the calculations, ensure that your calculator is set to handle decimals and powers correctly. Alternatively, you can use programming languages like Python or JavaScript to perform the calculations accurately and efficiently.