Fibonacci sequence empirical formula

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The discussion focuses on finding the empirical formula for the Fibonacci sequence, specifically the nth term, using the equation involving Phi (the golden ratio). The user attempts to apply the formula for n = 2 but encounters errors in expansion and simplification. The correct formula is given as Phin - (1 - Phi)n / √5, with Phi defined as (1 + √5) / 2. Assistance is requested to identify and rectify the mistakes in the calculations. Clarification on the expansion process is needed to ensure accurate results.
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Homework Statement


Research the Fibonacci sequence and hence find the empirical or explicit formula for generating the nth term of the fibonacci sequence. Use this formula to show that it does indeed produce the Fibonacci numbers for n = 1 to 5. You may not use calculators, expansions of phin should be done by expanding and simplifying in surd form.


Homework Equations


Phin-(1- Phi)n / \sqrt{}5

Phi = 1+\sqrt{}5 / 2



The Attempt at a Solution



For N = 2
substitute A into B
Answers is wrong when expanded and simplified
Any help would be greatly appreciated
Cheers.
 
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DavidGreen said:

Homework Statement


Research the Fibonacci sequence and hence find the empirical or explicit formula for generating the nth term of the fibonacci sequence. Use this formula to show that it does indeed produce the Fibonacci numbers for n = 1 to 5. You may not use calculators, expansions of phin should be done by expanding and simplifying in surd form.


Homework Equations


Phin-(1- Phi)n / \sqrt{}5

Phi = 1+\sqrt{}5 / 2



The Attempt at a Solution



For N = 2
substitute A into B
Answers is wrong when expanded and simplified
Any help would be greatly appreciated
Cheers.

It should work fine. You'll have to explain what's going wrong.
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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