Proof: Fibonacci Sequence Sums to Squares

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The discussion centers on proving that the sum of products of consecutive Fibonacci numbers equals the square of the last term in the sequence. The suggested approach is to utilize mathematical induction, leveraging the Fibonacci recursion formula for the inductive step. Participants express curiosity about whether alternative methods exist for the proof. The conversation clarifies that this is not a homework question. The focus remains on the mathematical proof rather than personal opinions.
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if a_1, a_2, a_3 .... belong to the fibonacci sequence, prove that

a_1a_2 + a_2a_3 + ... + a_{2n-1}a_{2n} = (a_{2n})^2
 
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If you are familiar with mathematical induction then that's the way to go with this one. Using the recursion equation (a_{2n} + a_{2n+1} = a_{2n+2} etc) should let you make the inductive step fairly easily.

BTW. Is this homework ?
 
not at all!
 
Are you familar with mathematical induction?
 
Is there any other method other than this?
 
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