I posted the question several days ago and Mathmate helped to guide me along. I am just unsure if I do not understand what he is saying or if I did complete the problem right. I would appreciate any input.


1.618034 - -.618034 = 2.236068/sqrt 5 = 1.00000

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.

Using Binet's formula,

F(20) = ((1+√5)^20 - (1-√5)^20)/(√5 * 2^20)

1+√5 = 3.236068
1-√5 = -1.236068

3.236068^20 = 1.5861811E10
1.236068^20 = 69.31819834

so, F(20) = 6765

To calculate the 20th Fibonacci number using the Binet formula, you will need to use the golden ratio values you have mentioned: 1.618034 and -.618034.

The Binet formula for the nth Fibonacci number is given by:
Fn = (phi^n - psi^n) / sqrt(5)

where phi is the golden ratio (1.618034) and psi is the conjugate (-.618034).

So, to find the 20th Fibonacci number, you can substitute the values into the formula:

F20 = (1.618034^20 - (-.618034)^20) / sqrt(5)

Now, to calculate this using a calculator, you can use the power function and the square root function. Here's how you can do it step by step:

1. First, calculate the individual powers of phi and psi:
phi^20 = 1.618034^20
psi^20 = (-.618034)^20

2. Next, calculate the subtracted terms:
phi^20 - psi^20

3. Finally, divide the subtracted terms by the square root of 5:
(F20 = (phi^20 - psi^20) / sqrt(5))

Now, when you calculate this using a calculator, you should get the result as the 20th Fibonacci number. In your case, if you obtained 6,765 as the result, that means you have correctly calculated the 20th Fibonacci number using the Binet formula.

In summary, to utilize the Binet formula to find the nth Fibonacci number, substitute the values of phi, psi, and n into the formula and then simplify the expression using a calculator.