what is the difference between Binet's formula to its simplified version? Are there any rules on when to apply which and can you show how the formula is condensed to the simplified version.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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what is the difference between Binet's formula to its simplified version? Are there any rules on when to apply which and can you show how the formula is condensed to the simplified version.

Binet's Formula is an explicit formula used to find the nth term of the Fibonacci
sequence. It is so named because it was derived by mathematician Jacques Philippe
Marie Binet, though it was already discovered by Abraham de Moivre. Based on the
golden ratio, Binet's formula can be represented in the following form:
If Fn is the nth Fibonacci number, then
√5
Fn
√/13 ((¹+₂✓ ³ ) " - ( ¹ = 2√³) ")
/5
2
SIMPLIFIED FORM:
$" -(-6)-"
Fn
√√5
(1 + √5)" - (1 - √5)"
2" √√5
Transcribed Image Text:Binet's Formula is an explicit formula used to find the nth term of the Fibonacci sequence. It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already discovered by Abraham de Moivre. Based on the golden ratio, Binet's formula can be represented in the following form: If Fn is the nth Fibonacci number, then √5 Fn √/13 ((¹+₂✓ ³ ) " - ( ¹ = 2√³) ") /5 2 SIMPLIFIED FORM: $" -(-6)-" Fn √√5 (1 + √5)" - (1 - √5)" 2" √√5
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