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Fibonacci sequence

The Appeal of 'Touch' -- Where The Fibonacci Sequence Connects Us All
The Fox Television series "Touch" indicates that there is, and it starts with the Fibonacci Sequence. In the television drama "Touch," actor Kiefer Sutherland portrays a man, Martin Bohm, who has been unwittingly chosen to facilitate the occurrence of ...

Book on Nature's Mysterious Fibonacci Sequence Featured on Fox's “Touch”
Sarah C. Campbell's children's book on the Fibonacci sequence is featured during the Fox TV show “Touch.” During the nationally-televised premier episode, one of the show's main characters uses “Growing Patterns” to learn more about the Fibonacci ...

Does Touch Get the Math Right?
Thematically, that's good, since that number does occur a lot in nature, often by way of its closely associated Fibonacci sequence. Which makes it all the more perplexing that, midway through the first episode, we have Danny Glover's character ...

Brigantine sculptor gets chance to show off his passion at home and school
The shape echoes nature in that it follows the Fibonacci sequence — 0, 1, 1, 2, 3, 5, 8, 13, and on and on. The last two numbers in the sequence are added together to get the next number. When plotted out on a graph the numbers spiral around each ...

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How can you figure out the sequence in this (Fibonacci) pattern? by Q: My teacher gave us this problem, and told us to solve it using Fibonacci. The sequence is a sequence that is neither arithmetic nor geometric, for which you are able to write a general term. _, 8, _, _, 27, _, ...

A: If it's a Fibonacci like sequence, then that means that each term (starting with the 3rd) is sum of previous two terms. So we have: x, 8, x+8, x+16, 2x+24, 3x+40, ... 5th term = 27 5th term = 2x+24 2x + 24 = 27 2x = 3 x = 1.5 So first 6 terms are: 1.5, 8, 9.5, 17.5, 27, 44.5, ...

how does the fibonacci sequence relate to Sunflowers? by Gymnastxoxo Q: i need it for a project and i dont really get it, the sequence is a spiral, and sunflowers are spirals?

A: See these Internet links :

What are the 31st, 32nd, 44th and 45th terms in the Fibonacci sequence? by smilinchick_21 Q: Also, what is the sum of the squares of F31 and F23? The sum of the squares F44+ F55? Is there a rule? I have to other rules but they aren't related to square numbers in the sequence.

A: F31: 1346269 F32: 2178309 F44: 701408733 F45: 1134903170 Sum of squares of F31 and F23: =1812440220361 + 821223649 =1813261444010 Sum of squares of F44 and F55: =491974210728665289 + 19483654655064681378025 =19484146629275410043314 I can't really think of a rule, though.

How on earth is the Fibonacci sequence shown in nature? by Q: When they say that it is nature's way of counting...or that it can be seen in leaves, shells, strawberry seeds, pineapple scales...how does it work? When I count the scales of a pineapple, the rows don't add up like the sequence does. Can anyone explain this to me?

A: 1)Layers of petals of some flowers. 2) The ratio of two consecutive fibonacci numbers is the golden ratio. You can read up on that.

What is the ratio for successive terms in the fibonacci sequence? by Q: If F4 is the fourth term in the sequence, therefore represents 3 and F5 is the fifth representing 5, what is F5/F4? What value does this ratio seem to be getting closer to as n (the 4 or 5) get larger? Use as many ratios as you need to see the long term value and investigate (Fn+1/Fn).

A: Here are the first twenty terms (and ratios) for the sequence: n Fn Ratio 1 1 2 1 1 3 2 2 4 3 1.5 5 5 1.666666667 6 8 1.6 7 13 1.625 8 21 1.615384615 9 34 1.619047619 10 55 1.617647059 11 89 1.618181818 12 144 1.617977528 13 233 1.618055556 14 377 1.618025751 15 610 1.618037135 16 987 1.618032787 17 1597 1.618034448 18 2584 1.618033813 19 4181 1.618034056 20 6765 1.618033963 The ratio is approaching (1 + √5)/2, which is the Golden ratio. A rectangle is said to have the most pleasing proportions if its height and width have this ratio. (1 + √5)/2 = 1.618033989

What is a good movie about the Fibonacci sequence? by Lauren Q: I need to find a documentary about Fibonacci and the Golden Ratio and stuff like that. Also, any movies that mention it or use it as part of the plot would be helpful.

A: The DaVinci Code

The fibonacci sequence or the golden ratio, something to base your life upon? by Prymer55 Q: I know this sequence and ratio can be directly related to life but should they govern how we should carry on with our lives? Not as a religion but maybe as a spiritual path? How could one follow such a path?

A: Zen is somewhat based on this. As any ratio indicates there is a balance in life. If you examine the food pyramid you'll see that the Golden ratio applies. It shows up in many of our activities. You can follow it by basing everything on it. Activities, time, locations, all can be made to conform to the ratio. There is another "golden ratio" called Parado's Principle. I have based much of my life on it and it generally holds true. The principle is simple -80/20. 80% is governed by 20%. For instance: 80% of the money is held by 20% of the people.

Why does the Fibonacci sequence never equal exactly the golden ratio when divided? by Q: What is the missing factor, variable, in the fibonnaci sequence to get exactly the golden ratio?

A: Ratio of two consecutive terms of the Fibonacci sequence is always rational (ratio of two integers), but the golden ratio [(1+√5)/2] is an irrational number. So they can never be equal. [But the ratio of two consecutive terms of the Fibonacci sequence tends to the golden ratio when n tends to infinity.]

What is a number sequence similar to Fibonacci? by thatonedudewhosaysstuff Q: I am teaching a programming course at my high school and I am kind of doing a number sequence unit right now. Last week I had them figure out how to make a program that generated a Fibonacci number sequence. I was wondering if anybody could recommend some other number sequences that would be a similar difficulty to code? btw this is in C++. Just what I was looking for, these answers have diversity and yet they are still all good. Thanks!

A: You could have the sequence consisting of: 1, 2, 2, 4, 8, 32, 256, 8192... that is the sequence defined by x_1 = 1 x_2 = 2 x_n = x_{n-1} * x_{n-2} or the Fibonacci sequence but with multiplication. This may be too easy since it is really just the same. Maybe this would be good for the test just as a "were you paying attention" check. There are *tons* of sequences over on the "On-Line Encyclopedia of Integer Sequences" but you'd have to pick some that have a nice computational property. One that you can use to talk about numbers that grow very large very quickly is the Ackermann function. This function is pretty easy to code, it is just two recursive calls, like Fibonacci, but it gets *crazy huge* even for fairly small input numbers. In fact, this function is the standard and semi-snotty retort whenever anyone argues about simple functions not being complex, or someone has a new compiler that you want to try and crush. Another would be to find perfect numbers. These are numbers where when you take half the sum of its proper divisors, you get the number itself. Examples are: 6 (1+2+3+6)/2, 28 (1+2+4+7+14+28)/2... You could even mention that it is an open question in mathematics whether or not there are any odd perfect numbers. Lastly, an interesting bunch of numbers are "friendly numbers." Two numbers are friendly if they share the same ratio of their divisors to the number itself: 6 and 28 are friendly because: (1+2+3+6)/6 = 2 and (1+2+4+7+14+28)/28 = 2 You can even reuse some of your code that you wrote for finding divisors of a number! Code reuse, what a great lesson.

Please help pseudo code for this particular Fibonacci sequence? by Q: Im trying to make a program in Visual basic that takes an initial number and a final number the initial number is where the Fibonacci sequence will begin and the final is where it will end, if someone could help me out with the logic please! pseudo code or VB Any help is appreciated thanks!

A: See this logic in Matlab. It's an almost direct-conversion to VB http://matrixlab-examples.com/fibonacci-numbers.html .

Does the Fibonacci sequence have to start with 0? by Jeph Q: Is a Fibonacci sequence the one that goes 0112358 or is a any sequence in that pattern a Fibonacci sequence? eg 2358; 31459; 347

A: must start with 0

Need ideas related to the Fibonacci Sequence and/or Leonardo Fibonacci.? by in bℓack && white Q: I am presenting a project in math class tomorrow over the famous mathematician Leonardo Fibonacci, focusing specifically on the Fibonacci Sequence. This is the equation for a perfect spiral and it's the one that goes "0,1,1,2,5,8,12...". I wanted to bake a food or even just buy something at the store that I can pass out tomorrow to the class that still ties into my project. Any suggestions? Even if it's just a design for decorating cookies? Thank you!

A: The Golden Ratio and Fibonacci triangle

What's the sumformula that connects pascal's triangle and the fibonacci sequence? by ann b Q: Like when you look at the sums of the diagonals of the triangle, you'll find the fibonacci sequence. what is the sumformula that describes these sums? Like the sumformula's answers on m=0 -> 1 (the top of the triangle, and first number of sequence). m=1 answer 1 m=2 answer 2 etc. (I am making an essay 'bout fibonacci and I think this is nice information to use).

A: Fn+1 = sum[C(n-k, k)] from k=0 to int(n/2)

Who created the fibonacci sequence? by BaBy PhAt LoVeR Q: 2. What is the fibonacci sequence? 3.How is the fibonacci sequence produced? 4.How does golden ratio relate to the fibonacci sequence? 5.Name two different ways the fibonacci sequence is found in nature If u noe the answer 2 any of these ?'s can u please answer them

A: Leonardo of Pisa known as Filius Bonacci (son of Bonacci). His father was an Italian merchant who traded in North Africa which is where the boy grew up and became exposed to Arabic mathematics. It was through contacts like this that Arabic and Indian mathematical ideas entered mainstream European culture, He was led to the sequence by considering the problem of the growth of a rabbit population and published his findings in his influential book Liber Abaci (Book of the Abacus) in 1202, He considered the growth of an idealised (biologically unrealistic) rabbit population, assuming that: (a) in the first month there is just one newly-born pair, (b) new-born pairs become fertile from their second month on (c) each month every fertile pair begets a new pair, and (d) the rabbits never die He argued thus: Let the population at month n be F(n). At this time, only rabbits who were alive at month n−2 are fertile and produce offspring, so F(n−2) pairs are added to the current population of F(n−1). Thus the total is F(n) = F(n−1) + F(n−2). (3) And that is the generating rule for the Fibonacci sequence. Each term is the sum of the previous two terms. (2) Conventionally it begins with 0, 1 and continues 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657 ... (4) The Golden Ratio, Phi 1.618033989 .., and its inverse phi (0.618033989 ...) are both produced by considering how the ratios of successive terms of the Fibonacci sequence converge (from either side) on the Golden Ratio. 3/5 = 0.6 5/8 = 0.625 8/13 = 0.61538 ... 13/21 = 0.61904 ... 21/34 = 0.61765 ... 34/55 = 0.61818 ... 55/89 = 0.61797 ... 89/144 = 0.61805 ... 10946/17711 = 0.618033990 ... 17711/28657 = 0.618033988 ... AND SIMILARLY 5/3 = 1.666 ... 8/5 = 1.6 13/8 = 1.625 21/13 = 1.61538 ... 17711/10946 = 1.618033985 ... 28657/17711 = 1.618033990 ... (5) The two best-known examples of Fibonacci numbers appearing in nature are in the arrangements of sunflower heads and pine cones.

What is the 1,976th number in the Fibonacci sequence? by williamsjw4 Q: I am trying to figure out what the 1,976th number in the Fibonacci sequence is...can anyone figure it out? If so, tell me how long ti takes you to figure it out or for your computer to solve the probem. It took my laptop 12 seconds to compute the 40th sequence number, so I imagine that it will take it 9 hours to compute the 1,976th number in the sequence.

A: I used a TI-86, but I'm guessing you would like to use a computer code to do this. For what it's worth, the TI-86 produces the result: 4.07466709145E412 using the algorithm: F=((((1+sqrt(5))/2)^1976)/sqrt(5))+.5 EDIT: whipping up a quick matlab script did not turn out to be fruitful but may get you started. This code reached an integer limit in Matlab I believe. EDIT2: OK, one more try. This code will produce the results you want with Matlab and the Symbolic Toolbox using the command 'vpa' (taken from link below) >> A=[0,1;1,1] >> vA = vpa(A,30) >> vA^10000 These are usefull resources: (check the link to the matlab group) http://www.congruentialuminaire.com/cLBlog/Lists/Posts/Post.aspx?ID=3 http://www.congruentialuminaire.com/cLBlog/Lists/Posts/Post.aspx?ID=5

how is the fibonacci sequence similar to pascal's triangle? by xoxo 03 Q: I'm a little confused on what Fibonacci is. I understand Pascal's. But obviously, I cannot see the connection between the two when I don't understand. So, what is the Fibonacci sequence and how is it related to pascals triangle? Thanks so much! Best answer will be given to who really deserves it!

A: The first two terms in the Fibonacci sequence are: 1, 1 Every subsequent term is formed by adding the two before it: 1, 1, 2 1, 1, 2, 3 1, 1, 2, 3, 5 1, 1, 2, 3, 5, 8 etc. This is similar to Pascal's triangle, where the outside is formed of 1s, and each internal cell is formed by adding the two above it.

Fibonacci Sequence appears in patterns connected with the golden ratio? by Jo Q: can anyone PLEASE explain this to me, how it works, or what the answer to this is? The Fibonacci Sequence appears in patterns connected with the golden ratio. Conjecture: when φ is raised to a positive integer power, the result can be written as A + Bφ where A and B are Fibonacci numbers. Find φ(2nd power), φ(3rd power), φ(4th power) to gather evidence for this conjecture. can you determine a pattern emerging from your calculations? thanks in advance

A: Okay, φ is defined to be the solution to: x^2 - x - 1 = 0 Hence φ^2 = φ + 1 So φ^3 = φ(φ + 1) = φ^2 + φ = 2φ + 1 φ^4 = 2φ^2 + φ = 2(φ + 1) + φ = 3φ + 2 ... Do you see where I'm going with this? It is clear that if we continue multiplying by φ and simplifying, we can always write φ^n as Aφ + B Well, what exactly are A and B? Well, since we have an inductive sort of method, it makes sense to work out what φ^n is in terms of smaller powers: φ^n = φ^(n-2) * φ^2 = (φ+1)φ^(n-2) = φ^(n-1) + φ^(n-2) So the nth power is the sum of the previous two powers. Does this surprise you? It is exactly how the Fibonacci sequence works! Therefore you should be able to see that A and B are in fact Fibonacci numbers in the sequence. Well, for n = 1, A = 0 and B = 1 for n = 2, A = 1 and B = 1 and using these initial conditions you can see that the A's will follow the sequence: 0, 1, 1, 2, 3, 5, ... And the B's will go: 1, 1, 2, 3, 5, ...

What's the General Term for the Fibonacci Sequence? Can anyone give me a site that teaches Number Sequences? by Fat Q: I need this TODAY. Please help me. Leonardo Fibonacci made this sequence, how do you solve it? How do find the next term? An?

A: T(n) = [ (phi ^ n) - (-1/phi)^n ] / sqrt 5 [ phi = (1 + sqrt 5) / 2 ] I dont think it can be derived, only can be proven through Mathematical Induction. i hope this helps

how is The Fibonacci Sequence displayed in nature? by olivia Q: i need to write a paper about the fibonacci sequence and what it is and stuff, ive never even heard of it before. so please help if you can, thankyou .

A: The numbers in the series are the sum of the previous 2 numbers, starting with 0 and 1. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc Link shows series with respect to plants

Links between fibonacci sequence and the golden ratio? by Answerme Q: I have 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13,...... ( i would call them something, but i dont know what they're called. But if your're smart, you will recognise them as part of fibonacci) And i am asked to comment on these findings and explain the link between the fibonacci sequence and the golden ratio. Could you please help me? What are the links? What do i comment on?

How does PHI (The divine proportion) relate to the Fibonacci sequence? by Petitefoxx Q: I am doing a paper on Fibonacci, and I noticed that in the book the Davinci code, the fibonacci sequence is related to PHI. Since the book is fiction, I was wondering if the two are really linked?

A: There is no intrinsic relationship as far as I know, and I did a paper on it in college. The Fibonacci Series Ratio is the music ratio, determining the relationships between musical instruments in the orchestra, although it has many other applications. Look up Isaac Asimov's book Science, Numbers, and I for a more detailed examination of the Fibonacci Ratio--Bry

What are some applications of the Fibonacci Sequence? by Oliver S Q: So I'm elaborating on the fibonacci sequence for a paper in my math class. I've found a lot of nickel and penny facts but i'd prefer to have some practical applications. Thanks :)

A: http://en.wikipedia.org/wiki/Fibonacci_number

If the Fibonacci Sequence doesn't start at 1, does it have to start with 2 repeating numbers? by kimpenn09 Q: I'm in a bit of an argument with someone. One of us says that there is only 1 Fibonacci sequence, it starts at one, and continues from there. One of us says that a Fibonacci sequence is simply a sequence of numbers where the first two numbers are the same (as in, 5, 5, 10, 15, 25, etc.)

A: There is only one Fibonacci sequence, beginning with 0 and 1. The sequence you are describing beginning with 5 and 5 is a true sequence, however, it is not the Fibonacci sequence.

Demostrate that Binets formula is valid by producing 6th term of fibonacci sequence? by Just another brick in the wall Q: Demonstrate that binets formula is valid by producing the 6th term of the fibonacci sequence (which is 8) Please, thanks.

A: The 6th term of the standard Fibonacci sequence = 5. Binet's formula is F_n = (1/sqrt5) (phi^n - psi^n). Hence for n = 5 F_5 =(1/sqrt5) [ ((1/2(1 + sqrt5)^5 - (1/2(1-sqrt5)^5)] F-5 = 5 as was to be shown Is that sufficient? Just be careful with the arithmetic. It's all a bit too long to type here but get back if you get stuck or can't get it to work out. Oh, just spotted what you said that the 6th term is 8. I make it the 7th term: The Fibonacci numbers are: 0,1,1,2,3,5,8,13, ...

what is the formula to find the nth term of the fibonacci sequence? by Melody Tai Q: what is the formula to find the nth term of the fibonacci sequence? the one with the square root and stuff. if anyone knows, pls tell me!! thank you!!

A: The formula is F(n) = (φ^n - (1 - φ)^n) / sqrt(5), where φ ("phi") is the golden ratio, (1 + sqrt(5)) / 2, which is approximately equal to 1.62.

How do I go about writing a Visual Basic program based on the Fibonacci Sequence? by Speakeasys Q: Write a Visual Basic program that will calculate the sum of all even numbers in a Fibonacci sequence that begins with 0, 1 and continues until the largest number in the sequence (NOT THE SUM) is greater than or equal to 2,000.

A: uh.. i think this is what you want.. Dim seq As New System.Collections.Generic.List(Of Integer) seq.Add(0) seq.Add(1) Dim going As Boolean = True While seq(seq.Count - 1) < 2000 seq.Add(seq(seq.Count - 1) + seq(seq.Count - 2)) End While Dim finalnumber As Integer = 0 For Each num As Integer In seq If (num Mod 2 = 0) Then finalnumber += num End If Next MsgBox(finalnumber) Assuming i understand the question.. finalnumber = 3382 the program adds 0,1 to the list.. and loops through 'till the last number in the sequence is >= 2000 while adding the two previous in the list. then in loops through the sequence and adds up the even numbers.

Is there a world record for the farthest number a person has gotten to in the Fibonacci Sequence? by Lujacris Q: I was just wondering if there was a world record for the farthest a person has gotten in the Fibonacci Sequence, and how far was it? Thanks!

A: I don't know, but there is a simple closed form for the Fibonacci numbers and the number of bitwise operations required does not grow very quickly so there would be few impediments to its calculation. http://upload.wikimedia.org/math/9/6/8/968be88f42e32712cb10d89a765ce708.png

Write a program to give the fibonacci sequence and how to handle exceptions.? by Q: Write a program to give the fibonacci sequence and how to handle exceptions. What is the time complexity of this and how to improve the program.

A: In what language? Here's one code in Matlab. It has to do with iterations, recursion, (exceptions?)

how is the Fibonacci sequence used in nature ? by Q: for my homework i have to do a presentation about the Fibonacci sequence and i need to know how its used in nature ?? HELP please need to know asap.

A: Plants form spiral to achieve efficient packaging or surface exposure to sunlight. The arrangement of seed in the composite head of a sunflower follow radiating spirals. Cones package seeds in a spiral of scales while stems spiral branches and leaves to avoid self-shading. Some 90% of plants show a Fibonacci phyllotaxis (leaf arrangement moving out along the stem) with the number of elements positioned being successive elements in the Fibonacci sequence. The spiral of leaves around a stem can follow a Fibonacci sequence of spacing as the stem elongates. Apical meristem cells, which will later develop into organs like leaves or flower petals usually form in the least crowded spot along the growing tip of a plant. The plant grows placing the second cell as far as possible from the first, and the third is placed at a distance farthest from both the first and the second cell in the bud. As the number of cells increases, the divergence angle between each successive cell eventually converges to a constant value of 137.5 degrees and thus creates Fibonacci spirals. The spirals are not a law of nature but are a fascinatingly prevalent tendency of cellular growth in a limited space. Fibonacci in nature http://goldennumber.net/nature.htm http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#plants

What codes do you use to make a program displaying a Fibonacci sequence when you run it? by carlo_nomad16 Q: an example of a Fibonacci sequence is: 1,1,2,3,5,8,13,21,34,55,89... the logic behind it is: 1+1=2, 1+2=3, 2+3=5, and so on.

A: a=1 b=1 print a print b s=0 while (some break condition) begin ... s=a+b a=b b=s print b ... end

How many numbers are in the Fibonacci sequence? by Q: I was in school and we started talking about the fibonacci sequence, and my teacher said its extra credit for the first one who turns in how many number in it! Help?! :)

A: There are infinitely many terms. Each term is the sum of the two previous terms, so there never is any last one.

what is the formula for the nth term of the Fibonacci sequence? by ~*Smiley Girl*~ Q: I searched on google and it came up with quite a few things. Also, could you please give me some examples of sequences similar to the fibonacci sequence? Thanks! :)

A: It is called Binet's formula. You can find it here: http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html Any sequence whose n-th term is computed based on the previous two terms might be considered 'similar' to the Fibonacci sequence. For example: 1, 2, 4, 8, 16, 32, 64, ... Here the n-th term is 2 times the (n-2)-th term, plus the (n-1)-th term. Another example: 0, 3, 3, 6, 9, 15, 24, 39, ... Here, like the Fibonacci sequence, each term is the sum of its two closest predecessors.

How do you use the equation for a fibonacci sequence? by dennis r Q: the equation for the fibonacci sequence is F(n)=F(n-1)+F(n-2), when n>1. Yet when i try to solve for known numbers in the sequence i don't get the next number. what am i doing wrong? n is the variable and F would be the next number in the sequence right?

A: F(n)=F(n-1)+F(n-2), where F(0) = 0 and F(1) = 1 n= 2 F(2) = F(2-1) + F(2-2) F(2) = F(1) + F(0) F(2) = 1 + 0 = 1 n = 3 F(3) = F(3-1) + F(3-2) F(3) = F(2) + F(1) F(3) = 1 + 1 F(3) = 2 n = 4 F(4) = F(4-1) + F(4-2) F(4) = F(3) + F(2) F(4) = 2 + 1 F(4) = 3 n = 5 F(5) = F(4) + F(3) F(5) = 3 + 2 F(5) = 5

What is the 100th term in the Fibonacci Sequence? by mark_10_4 Q: I need to write a paper about the Fibonacci sequence for Algebra and one of the requirements is to write down the 100th term in the Fibonacci Sequence.

A: I didn't do the calculations, but I searched on it and found several sites with all sorts of info. 100th term is 354224848179261915075 ----------------------------------------------------

What are 5 interesting facts about the Fibonacci Sequence? by Q: I have a project due this week and i need 5 facts on the Fibonacci Sequence. Help!

A: Do a Web search on "Fibonacci" and you will find tons of sites about the Fibonacci sequence, trust me. Then just pick five facts you like.

How to make a program for fibonacci sequence in java? by Q: We need to do this for school, and I have no idea how. "subsequent coding for a program which finds the nth term of a Fibonacci sequence starting from 1. You must do so using if and else if statements and do while statements somewhere within your coding. The maximum number of terms permitted should be between 1 and 50 (otherwise they get too big!)" I've found other people's ones, but none with do while in them

A: The Fibonacci sequence begins with 1, and the second term is 1, and each term after that is the sum of the two before it. So the first five terms of the FIbonacci sequence would be 1, 1, 2, 3, 5, for example. It is possible to write this recursively, but recursion doesn't use a while (Recursion takes a problem and breaks it up into an "exit condition" (like "if n == 1 return 1; else if n == 2 return 1)" e.g. conditions that make it terminate, and also one or more larger conditions that break the problem down into smaller pieces that are guaranteed to eventually lead to an exit condition, like "... else return fibonacci( n-1 ) + fibonacci( n-2 )" ) To use a while loop saves you the runtime and stack frames that recursion would cost you. This would be the "iterative" version (which is what uses a while). Note that to do this, you have to have it "remember" the previous two terms. The approach to write the iterative version is similar. You want to think about what will make your loop exit, and how to start off. Since the assignment tells you the maximum terms permitted should be between 1 and 50 (I assume inclusive), I interpret that to mean they want you to check for that, yourself. So you could begin with something like "if( n >= 1 && n <= 50 ) { /* entire fibo routine here */ }" public class CheckFibo {   public static void fibo ( int n ) {     if( n >= 1 && n <= 50 ) {       int prev1 = 1; int prev2 = 1;            // previous term and one before that       int i = 3;                          // counts up terms; "i" will start at the 3rd term       if( n > 0 )                          // print out first term, as long as at least one term requested       System.out.print( prev2 + " " );        if( n > 1 )                          // print out second term, if requested       System.out.print( prev1 + " " );        while( i <= n ) {                      // compute and print remaining terms, if any         int curr = prev1 + prev2;            // calculate current term         System.out.print( curr + " " );          // output it         prev2 = prev1;                    // prev term now becomes penultimate term         prev1 = curr;                      // term just computed now becomes "prev" term         i++;                             // ith term just printed, so increment i       }     }   }     public static void main( String[] args ) {     int n = 7;                            // print n terms of Fibonacci series     System.out.println( "printing first " + n + " terms of Fibonacci series" );     fibo( n );   } }

How to come up with the formula of the Fibonacci Sequence? by Bill M Q: The fibonacci sequence fn is defined recursively by the conditions f1 = 1, f2 = 1 and fn = fn−1 +fn−2 for n greater than or equal to 3. How would I find the formula of fn.

A: ____________________ F(1)=1, F(2)=1, F(n+2)=F(n+1)+F(n), n≧1 _____________________ x^2=x+1, x^2-x-1=0 A=[(1+√5)/2], B=[(1-√5)/2]  A+B=1  AB=-1 F(n+2)-A・F(n+1)=B[F(n+1)-A・F(n)] F(n+2)-B・F(n+1)=A[F(n+1)-B・F(n)] because,   F(n+2)-(A+B)F(n+1)+AB=0   F(n+2)-F(n+1)-F(n)=0   F(n+2)=F(n+1)+F(n) __________________________________________ F(n+1)-A・F(n)={F(2)-A・F(1)}・B^(n-1) F(n+1)-B・F(n)={F(2)-B・F(1)}・A^(n-1) F(2)-A・F(1) =1-[(1+√5)/2]=[(1-√5)/2]=B F(2)-B・F(1) =1-[(1-√5)/2]=[(1+√5)/2]=A   F(n+1)-A・F(n)=B^n   F(n+1)-B・F(n)=A^n   (A-B)・F(n)=A^n - B^n A-B=[(1+√5)/2]-[(1-√5)/2]=√5 F(n)=[A^n-B^n]/(A-B)   in this case , the solution : F(n)=[{(1+√5)/2}^n - {(1-√5)/2}^n]/√5 _______________________ a bit of verification , F(1) =[{(1+√5)/2}- {(1-√5)/2}]/√5 =√5/√5=1 F(2) =[{(1+√5)/2}^2 - {(1-√5)/2}^2]/√5 =[(6+2√5)-(6-2√5)]/(4√5) =(4√5)/(4√5)=1 F(3) =[{(1+√5)/2}^3- {(1-√5)/2}^3]/√5 =[(1+3√5+15+5√5)-(1-3√5+15-5√5)]/(8√5) =(16√5)/(8√5)=2 . ______________

What is a good hands-on demonstration for math class involving the Fibonacci sequence? by √ξяǿйٲ¢Д Q: I need to write a paper and do an in-class presentation for my high school pre-calculus class. It can be on any mathematical or real-life, math-related subject that I choose. I would like to do the Fibonacci sequence, because it sounds really interesting and there's a lot of math related to it. However, I need to include a hands-on demonstration in my presentation, and I'm having trouble coming up with a demonstration involving the Fibonacci sequence. Ideas? Any demonstration ideas involving other topics would be helpful too. Thanks :)

A: Have you considered an investigation into whether sequences similar to the main Fibonacci one behave in the same way. Try sequences that have the same rule, i.e. each term the sum of the previous two, but that don't start with 1, 1, e.g. 2, 7, 9, 16, 25, 41, 76, . . . -5, -1, -6, -7, -13, -20, . . . Does each term divided by the previous one work towards a limit like the golden number for the main Fibonacci sequence? You might be surprised by the answer. You can go even more weird. What about a sequence where each term is the sum of the one two before and the one three before. You will need to start with three terms. In the one below I made up 2, 5, 6 to start with. 2, 5, 6, 7, 11, 13, 18, 24, 31, 42, . . . Then there are the practical applications of Fibonacci sequences such as the prongs on a pine-cone. Type "Fibonacci sequence" into google and you will get far more than you need. Yes, it is a fascinating topic.

What is the formula for the nth term of the Fibonacci sequence? by ~*Smiley Girl*~ Q: I searched on google and it came up with quite a few things. Also, could you please give me some examples of sequences similar to the fibonacci sequence? Thanks! :)

A: Try WIki http://en.wikipedia.org/wiki/Fibonacci_number

How can I solve a Fibonacci sequence question by a faster way? by Shirley Q: For example, if I want to find the 15th term of the Fibonacci sequence, how can I solve it by a faster way?

A: Use Binet's formula.

What is the pattern between the area sums and the original fibonacci sequence? by luckyDOG Q: In the Fibonacci numbers, what is the pattern between the area sums and the original fibonacci sequence? It also said " Hint: look at the factors of the sums" Help please?

A: The area sums is based on starting with 1, then take the longer side (1) and make a square which will be 1 + 1 = 2, and the long side = 2, add a square 2 x 2 to the rectangle 1 x 1, the resulting longer side = 3, so add a 3 x 3 square, and the longer resulting side would be 5, so add a square 5 x 5 and so on, and so on......

Why do you think the Fibonacci sequence is emulated throughout the natural world? by Q: How many times do we see it? How could Fibonacci have known or seen this pattern? If the Fibonacci sequence repeats itself over and over again, what would that imply about so called "coincidences" in our day-to-day experience? @ Tim Actually, I've known about the Fibonacci sequence for a while, the pilot for "Touch" just brought it to my mind again. I in no way hold a belief in numerology, I just wanted to know its role in nature.

A: Never exactly. The very simple Fibonacci sequence has a repetitive fractal like growth, meaning simply that each step builds on a reflection of the proportions of the previous steps without the need for additional information. This is similar to some natural growths -- particularly growths that follow the path of least resistance or evolved to have the strongest, most efficient shell structure. The path of least resistance. The natural world never has an exact pattern though -- there are always exterior factors. It makes perfect sense, it's not mystical or anything. > How could Fibonacci have known or seen this pattern? Uh, he just added the two previous numbers. 0, 1, 1, 2, 3, 5, 8, 13. It's pretty simple really. > "coincidences" in our day-to-day experience? What coincidences? Repetitive growth is natural, the path of least resistance. IMO numerology is just the failure of inductive reasoning. Nothing magical. I'm guess you saw the Touch pilot with Kiefer Sutherland? These themes were really popular in indie films in the 90's. Pi directed by Daren Arronofsky in particular deals with this stuff pretty heavy-handedly. Pi (1998) http://www.imdb.com/title/tt0138704/ I also saw a BBC documentary recently on math in the natural world you might like. It was pretty good, scientific. BBC The Code http://www.bbc.co.uk/tv/features/code/

Where does the fibonacci sequence appear in nature? by ^________^ Q: Please answer my question! I have been wondering where the FIbonacci sequence appear in nature? I've asked if animals have anything to do with the sequence, and so far, my answers have been shot down! One of the people who answered told me that: animals have nothing to do with this sequence, so my only question after that was: Where? Where does this sequence appear in nature, if not in animals?

A: The Fibonacci number appears in any spiral pattern on plants - such as the arrangement of seeds in a sunflower or a pinecone, or the petals in a budding flower. I'm reading now that Fibonacci is present in spiral design such as the inner ear of mammals and the Nautilus shell, and the relative lengths of the human phalanges also follow Fibonacci's first numbers.

What is the sequence when taking 3 terms in the fibonacci sequence & multiplying 1st and 3rd together? by Q: After multiplying the first and last terms in the 3 numbers of the fibonacci sequence and subtracting their product by the square of the middle number I get 1 as the difference, but sometimes the subtraction is the opposite, is there a special sequence ?

A: If you are taking the sequence as 1, 1, 2, 3, 5, 8, 13, ......... with the first term 1, the second term 1, the third term 2, and so on, it appears that if the terms you multiply are odd numbered terms (subtracting the square of an even term) the result is +1. If the terms you are multiplying are even numbered terms (subtracting the square of an odd term) the result is - 1. This is based on inductive reasoning only, and in no way constitutes a proof.

What purpose does the Fibonacci sequence serve? by Michael C Q: The following is part of the Fibonacci sequence. 1123581321345589144 What purpose, if any, does this serve and is there any significance to the 12th number being 144? Since 144 is the product of 12 x 12. I am no mathematician and I may have the sequence all wrong.

A: there are a lot of applications of that Fibonacci sequence in mathematics, one of my favorites is the binomial theorem, the pascal's triangle in which that sequence is related. It's hard to enumerate everything, but you could read some articles about it and you'll be amazed on how it is used. About the 144 thing, im not sure if it has something to do with the 12x12, i just know it's the sum of 55 and 89 that's why it is there. ^_^

Why is the Fibonacci sequence important to mathematics? by Q: Why is the Fibonacci sequence important to mathematics? I need a direct answer.

How is the Fibonacci Sequence related to architecture? by echo Q: please don't tell me what the fibonacci sequence is. i already know.

A: The Fibonacci has a number of beautiful mathematical properties in geometry and can be found in famous monuments like the angle of the egypthian pyramides, Stonehenge, churges and monasteries - although neither fractional nor irrational numbers were not known until the mid-ages. http://www.antifool.com/cc/?c=fibonacci Leonardo Fibonacci was born in Pisa in the 12th century. He was a merchant and customs officer of the time, traveling widely in North Africa. He was also one of the first Europeans to learn about the Arabic numbers 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 and to persuade other people to use them; before then everybody counted in 12's. Leonardo was trying to find a way of modelling the population of rabbits. Let us suppose that any new pair of rabbits produces one pair in the next breeding season and one in the season after that, and then they die. This means that the total number of new pairs in a given season is equal to the number of new pairs born in the previous season, plus the number born in the season before that. So to find the next number in the sequence you add together the last number and the one before it. Starting with one pair of rabbits, you can easily generate the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... - the population of rabbits grows very quickly - actually exponentially fast. The surprising thing about Fibonacci's sequence is that it turns out to occur in many different places in nature. The way in which the spiral patterns of sunflower seeds and pine cones grow is described by the sequence, and it is common for the number of petals on a flower to be a Fibonacci number. Four-leaved clovers are rarer than five-leaved ones because five is in Fibonacci's sequence and four isn't.

fibonacci sequence? by Justin Park Q: is the fibonacci sequence never ending or does it just stop at: 1, 1, 2, 3, 5, 8, 13, 21?

A: All sequences are never ending, especially if they rely on previous terms.

Fibonacci Sequence? ? by Q: I was watching something, and in terms of the Fibonacci Sequence and the Golden Ratio, apparently people we find attractive have something to do with the Golden Ratio or the Fibonacci Sequence, please could you fully explain it, or even parts of it, thanks a lot:)

A: The golden ratio is a mathematical constant that occurs when the ratio between two numbers is equal to the ratio between the sum of the two numbers and the larger of the two numbers. If a > b. The ratio of a : b is equal to the ratio of a+b : a. This occurs when a is 1.61803398874989 times bigger than b. Therefore if a = 1.61803398874989b, then a+b = 1.61803398874989a The golden ratio is connected to the fibonnacci sequence. I'm not sure how. The fibonacci sequence is a sequence of numbers formed by adding together the two previous numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... A lot of people think that people find people attractive when different parts of their body are in the golden ratio. I'm pretty sure this is bullshit. I'm not aware of any research that has suggested a correlation betwen the golden ratio and attractiveness. However the golden ratio does appear in a lot of places in nature. Some research suggests there are structures on an atomic scale that exhibit the golden ratio.

Fibonacci sequence? by HAHAHA Q: I can not understand 2 questions about Fibonacci sequence. Please answer and explain clearly :D The Fibonacci sequence consists of the pattern 1,1,2,3,5,8,13,,, 1. Using your calculator, look at the successive ratios of one term to the next. Make a conjecture. 2. List the first eight terms of the sequence formed by finding the differences of successive terms in the Fibonacci sequence.

A: seems pretty easy lts look at the basics again That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers (sequence A000045 in OEIS), also denoted as Fn, for n = 0, 1, … , are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 in this example 0 plus 1 is 1, 1 plus 1 is 2 , 1 plus 2 is 3, 2 plus 3 is 5. no I wont do you home work for you you do it there is your answer if your THINK about it

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