Question
The Fibonacci numbers may be computed with the formula$$f_{k}=\frac{(1+\sqrt{5})^{k}-(1-\sqrt{5})^{k}}{2^{k} \sqrt{5}}$$Use this formula to compute $f_{1}, f_{2}, f_{3}, f_{4}$, and $f_{5}$. (Imagine computing $f_{100} .$ )
Step 1
Step 1: We are given the formula for the Fibonacci sequence as: $$f_{k}=\frac{(1+\sqrt{5})^{k}-(1-\sqrt{5})^{k}}{2^{k} \sqrt{5}}$$ We are asked to compute $f_{1}, f_{2}, f_{3}, f_{4}$, and $f_{5}$. Show more…
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Key Concepts
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An explicit formula for the $n$ th term of the Fibonacci sequence is $$F_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$ Apply algebra (not your calculator) to find the first two terms of this sequence and verify that these are indeed the first two terms of the Fibonacci sequence.
Sequences and Series
Fibonacci Sequence The famous Fibonacci sequence $\left\{u_{n}\right\}$ is defined recursively as $$ u_{1}=1 \quad u_{2}=1 \quad u_{n+2}=u_{n}+u_{n+1} \quad n \geq 1 $$ (a) Write the first eight terms of the Fibonacci sequence. (b) Verify that the $n$ th term is given by $$ u_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}} $$ (Hint: Show that $u_{1}=1, u_{2}=1,$ and $\left.u_{n+2}=u_{n+1}+u_{n} .\right)$
Infinite Series
Sequences
Define the sequence $$F_{n}=\frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right)$$ a) Find the first 10 terms of this sequence and compare them to Fibonacci numbers. b) Show that $3 \pm \sqrt{5}=\frac{(1 \pm \sqrt{5})^{2}}{2}$ c) Use the result in b) to verify that $F_{n}$ satisfies the recursive definition of Fibonacci sequences.
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