Extension: Fibonacci Sequence
Read this topicExplanations 6
- The Fibonacci Sequence is a special type of sequence over the positive integers, where the nnnth Fibonacci number is usually denoted FnF_nFn. The definition of the Fibonacci se…
By Daniel Liu, Catherine Yeo, Ruiran Xun
- Prove that Fn=⌊φn5⌋F_n=\left\lfloor\dfrac{\varphi^n}{\sqrt5}\right\rfloorFn=⌊5φn⌋ where ⌊x⌋\lfloor x\rfloor⌊x⌋ is the floor of xxx. Can you design an algorithm to find nnn wh…
By amaterasu, Moor Xu
- Use this exercise along with φm+n=φmφn\varphi^{m+n}=\varphi^m\varphi^nφm+n=φmφn to prove that Fm+n=FmFn+1+Fm−1FnF_{m+n}=F_mF_{n+1}+F_{m-1}F_nFm+n=FmFn+1+Fm−1Fn.
By amaterasu
- Prove that φn=φFn+Fn−1\varphi^n=\varphi F_n+F_{n-1}φn=φFn+Fn−1 where φ\varphiφ is the golden ratio.
By amaterasu
- Here's a really cool example of the Fibonacci sequence in nature!
By Expii Solve
- (Video) Intro to Series and Sequences by Eddie Woo Here is a fantastic video by Eddie Woo where he goes over the basics of sequences and series. In this video he goes over: ∘ Se…
By Amy Feraco
Problems 11
- We can say a lot about the sum ∑i=0nFi\sum\limits_{i=0}^nF_ii=0∑nFi. What about the following sum? Does it converge or diverge? ∑i≥01Fi\sum_{i\geq0}\dfrac1{F_i}i≥0∑Fi1
- What is the next term in the sequence 0,1,1,2,3,5,8,13,...0,1,1,2,3,5,8,13,...0,1,1,2,3,5,8,13,...?
- Suppose a1=2a_{1} = 2a1=2 and a2=3a_{2} = 3a2=3. We have a sequence that is described by an=an−1+2an−2a_n = a_{n-1} + 2a_{n-2}an=an−1+2an−2. What is a5a_{5}a5?
- Let's try a sequence similar to the Fibonacci numbers! If a0=0a_0 = 0a0=0, a1=1a_1 = 1a1=1, and for n≥2n \ge 2n≥2, an=2(an−1)+an−2a_n = 2(a_{n-1}) + a_{n-2}an=2(an−1)+an−2,…
- The Fibonacci sequence is defined by F1=1F_1 = 1F1=1, F2=1F_2 = 1F2=1, and Fn+2=Fn+1+FnF_{n+2} = F_{n+1} + F_nFn+2=Fn+1+Fn. What is F10F_{10}F10?
- The Fibonacci numbers are defined recursively by the equation Fn=Fn−1+Fn−2F_n = F_{n - 1} + F_{n - 2}Fn=Fn−1+Fn−2 for every integer n≥2n \ge 2n≥2, with initial values F0=0F_0…
- Let f1f_1f1, f2f_2f2, f3f_3f3, ..., be a sequence of numbers such that fn=fn−1+fn−2f_n = f_{n - 1} + f_{n - 2}fn=fn−1+fn−2 for every integer n≥3n \ge 3n≥3. If f7=83f_7 = 8…
- Mathematical objects appear everywhere, from manmade architecture to nature. Spring is here, and plants are flowering again! Which mathematical object appears in ferns, sunflowe…
- Did you know, that in nature, many plants have flowers with a number of petals that is a Fibonacci Number? Those are numbers from the sequence 1,2,3,5,8,…1, 2, 3, 5, 8, \dots1,2…
- You realize that you’ve had a serious problem with buying pumpkins this month. At first you had 000, and then every time you’ve been to the store you’ve bought: 1,1,2,3,5,8,13…1…
- Find the Zeckendorf decomposition of 287287287.