Fibonacci sequence empirical formula

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SUMMARY

The discussion focuses on deriving the empirical formula for the Fibonacci sequence, specifically the formula for the nth term represented as Φ^n - (1 - Φ)^n / √5, where Φ equals (1 + √5) / 2. Participants confirm that this formula should yield Fibonacci numbers for n = 1 to 5. However, one user reports an error in their calculations when substituting values, indicating a need for clarification on the expansion and simplification process.

PREREQUISITES
  • Understanding of the Fibonacci sequence and its properties
  • Knowledge of surd form and simplification techniques
  • Familiarity with the golden ratio (Φ) and its mathematical significance
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the derivation of the Fibonacci sequence using Binet's formula
  • Practice expanding and simplifying expressions in surd form
  • Explore the mathematical properties of the golden ratio (Φ)
  • Investigate common errors in algebraic substitutions and simplifications
USEFUL FOR

Students studying mathematics, particularly those focused on sequences and series, as well as educators seeking to clarify the Fibonacci sequence's empirical formula.

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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.
 

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