Consider a sequence defined similarly to the fibonacci but...

Name: consider a sequence defined similarly to the fibonacci but with a slight twist fn fn 1 fn 2 with f1 2 and f 2 5 generate terms f 3 f4 f5 f6 f7 and f8 then determine the value of f25 12482
Uploaded: 2022-06-08T02:44:11-08:00
Duration: 2 min 50 s
Channel: Michael Anderson
Description: consider a sequence defined similarly to the fibonacci but with a slight twist fn fn 1 fn 2 with f1 2 and f 2 5 generate terms f 3 f4 f5 f6 f7 and f8 then determine the value of f25 12482

Consider a sequence defined similarly to the Fibonacci, but with a slight twist: f(n) = f(n-1)-f(n-2) with f(1) = 2 and f (2) = 5 Generate terms f (3), f(4), f(5), f(6), f(7), and f(8). Then, determine the value of f(25).

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Involve the Fibonacci sequence, which is de- fined recursively as follows: $$F_{1}=1 ; \quad F_{2}=1 ; \quad F_{n+2}=F_{n}+F_{n+1} \quad \text { for } n \geq 1$$ (a) Complete the following table for the first ten terms of the Fibonacci sequence. $$\begin{array}{cccccccccc}F_{1} & F_{2} & F_{3} & F_{4} & F_{5} & F_{6} & F_{7} & F_{8} & F_{9} & F_{10} \\\hline 1 & 1 & & & & & & & & \\\hline\end{array}$$

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The sequence defined recursively by setting $$a_{\mathrm{n}+2}=a_{n+1}+a_{n} \quad \text { starting with } \quad a_{1}=a_{2}=1$$,is called the Fibonacci sequence. (a) Calculate $a_{3}, a_{4}, \cdots, a_{10}$ (b) Define $$r_{n}=\frac{a_{n+1}}{a_{n}}$$,$$\text { Calculate } r_{1}, r_{2}, \cdots, r_{6}$$, (c) Assume that $r_{n} \rightarrow L$, and find $L$. HINT: Rclatc $r$, to $r_{n}, 1$.

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$an-explicit-formula-for-the-n-th-term-of-the-fibonacci-sequence-is-f_nfrac1sqrt5n-1-sqrt5n2n-sqrt5-a$

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.

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00:02 Okay, consider a sequence defined similarly to the fibonacci sequence, but the slight twist.

00:07 So they give you the function f of n equals f of n minus 1 times or minus, oops, so minus f of f of n minus 2.

00:23 And then they give you f of 1 equals 2 and f of 2 equals 5.

00:29 So they want you to find f of 3, which equals, um, f of 3 minus 1 minus f of 3 minus 2.

00:43 3 minus 2, so that's 5 minus 2, which equals 3...

Question

Consider a sequence defined similarly to the Fibonacci, but with a slight twist: f(n) = f(n-1)-f(n-2) with f(1) = 2 and f (2) = 5 Generate terms f (3), f(4), f(5), f(6), f(7), and f(8). Then, determine the value of f(25).

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