Proving Fibonacci Numbers Equation f_2n+1=(f_n+1)^2+(f_n)^2

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SUMMARY

The discussion focuses on proving the Fibonacci numbers equation f2n+1 = (fn + 1)2 + (fn)2. Participants explore various approaches, including substituting values into the Fibonacci formula and considering mathematical induction as a potential method for proof. The equation fn2 = f2n - 2fn-1fn is also referenced, indicating its relevance in the proof process. The discussion highlights the complexity of the problem and the need for a structured approach to arrive at a solution.

PREREQUISITES
  • Understanding of Fibonacci sequence properties
  • Familiarity with mathematical induction techniques
  • Knowledge of algebraic manipulation of equations
  • Ability to work with recursive formulas
NEXT STEPS
  • Study the principles of mathematical induction in depth
  • Learn about the properties of Fibonacci numbers and their identities
  • Explore algebraic techniques for manipulating recursive sequences
  • Review examples of proofs involving Fibonacci sequences
USEFUL FOR

Mathematics students, educators, and anyone interested in number theory or recursive sequences will benefit from this discussion.

charity4thep
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Homework Statement


Prove that f_2n+1=(f_n+1)^2+(f_n)^2​


Homework Equations



(f_n)^2=(f_2n)-2(f_n-1)(f_n)

The Attempt at a Solution



i started with plugging in for the formula in (f_n)^2 and doing the same for (f_n+1)^2

but I am not sure if I'm even going in the right direction

i got to the point (f_2n+1) = f_2n+2 + f_2n -2f_n(f_n+1 + f_n-1)
 
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how about using induction... (not that I've tried yet, but worth a shot)
 

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