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(Solved): Recall that the Fibonacci sequence {Xk} can be defined as the solution of the second-order diffe ...
![(c) Show that
\[
\lim _{k \rightarrow+\infty} \frac{X_{k+1}}{X_{k}}=\frac{1+\sqrt{5}}{2}
\]
NOTE: The number \( \frac{1+\sqrt](https://media.cheggcdn.com/study/564/5645324b-fda2-49f9-95cb-10fabb1d13ce/image)
Recall that the Fibonacci sequence can be defined as the solution of the second-order difference equation: (a) Show that for any . (b) Show that for any .
(c) Show that NOTE: The number is known as the "golden ratio".
Expert Answer
Start with the definition of the Fibonacci ?k=1nXk.Replace each term X_{k} in the sum with the definition from step 1: ?k=1n(Xk?1+Xk?2).?k=1nXk=Xn+1+X