Fibonacci Sequence?

Question

I need some solid BASIC info on the Fibonacci sequence without its formula ( coz i didnt understand it) and its relation to the Golden Ratio...

Answer 1

first, start with 1 and 1 (1st and 2nd terms)

1+1 = 2

now, take te 2nd 1 (2nd term) and 2 (3rd term):

1+2 = 3

now take 2 (3rd term) and 3 (4th term)

2+3 = 5

This would produce all values for the fibonacci sequence: 1,1,2,3,5,8,13,21,...

Now, take the 1st 2 terms, and divide the 2nd by the 1st:

1/1 = 1

taking the 3rd and 2nd terms, and dividing the 3rd by the 2nd:

2/1 = 2

following the order:

3/2 = 1.5
5/3 = 1.67
8/5 = 1.625

Again continuing the divisions, we would get closer and closer to the golden ratio, 1.615 (to 4 decimal places). The ratio is actually an irrational number, like pi.

Answer 2

At the risk of doing your homework for you, here I go...

The Fibonacci sequence is easy enough to understand:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ...
Each number is the sum of the two numbers just before it. This has some pretty interesting applications in understanding the universe, but let's just leave that alone for now.

The Golden Ratio is also pretty easy to understand(-ish). The Ratio is 1:1.618034(ish). It's an -irrational- number and goes on and on and on.

The ratio is when two qauntites are related in this manner mathematically:

((a+b)/a) = (a/b) = phi (the greek letter representing the golden raio)

Again the golden ratio has some really nifty consequences in understanding art, architecture, nature etc., but lets just leave that alone for now.

You wanted to know HOW they're related. Lets take two numbers from the fibonnacci sequence above: 55 and 89. And lets divide 89/55. We get: 1.6181818... Hey, thats real close to the golden ratio isn't it? How about another. lets try bigger numbers like 610 and 377. 610/377= 1.618037.

What's going on here?

Lets look at the golden ratio formula with these numbers and see what happens:
Is this equation true? Does:
((610+377)/610)=(610/377)?
Let's see:
987/610 = 610/377? (Hey, 987 is next in the fib sequence!)
1.6180327 = 1.6180371, (well, it doesn't jive exactly but when we round to four significat digits...
1.618=1.618

Whenever you perform a calucation in the golden ratio you will end up with numbers from the Fibonacci sequence or in a fibonacci-type progression sequence. And the further you get in the Fibonacci sequence, the closer you get to the golden ratio.

It's a little bit more complex than that, but that's about as basic as you get when dealing with these formulas that are a lot more complex than they first appear.

Answer 3

Without the formula, Fibonacci came up with this sequence to examine the population growth of pairs of rabbits.

With the formula, anyway, the first two terms in the sequence are defined to each be 1, then every term afterward is the sum of the prior two.
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13, and so on.

The first handful of terms are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, and so on.

As far as its relationship to the golden ratio is concerned, examine any two consecutive terms in the sequence. The farther out you go, the close the ratio of those two terms will be to the Golden Ratio. As examples (rounded):

5 : 3 = 1.666666667
8 : 5 = 1.6
13 / 8 = 1.625
21 / 13 = 1.615384615
34 / 21 = 1.619047619, and so on. These ratios will come closer and closer to phi.

Go out even farther and you're closer to phi:
17711 / 10946 = 1.618033985
2178309 / 1346269 = 1.618033989

Answer 4

All I know is that the Fibonacci sequence is 1 1 2 3 5 8 13 21. Each number is the sum of the two numbers before it.

Answer 5

It is quite easy, in fact.

Put 0. That is Fibonacci first number.
Then put 1. Fibonacci second number.
Now add the previous two numbers together. 0 + 1 is 1. That is Fibonacci third number.
Then add the previous two numbers, 1 + 1 = 2. That is Fibonacci 4th number.
Then add the previous two numbers (that is #3 and #4) to get Fibonacci fifth number (1+2=3).
Then add the previous two numbers (that is #4 and #5) to get Fibonacci 6th number (2+3=5).
Repeat until the end of time.

Now, take two adjacent Fibonacci numbers, and divide the largest one by the otehr one. For instrance, 5/3. The result is 1.666666... Do that again with a larger pair, 13/8 = 1.625. Do that with a much larger pair, say 2584/1597 (#17 and #18), and you get 1.6180338... Much larger pair will give you a pregressively more accurate estimation of the Golden Ratio.

Answer 6

It is an interesting question to ask which sequences a_n have the property that each term (after the first two) is the sum of the previous two terms. Of course, these sequences will only depend on the first two terms, so there's kind of a "two-dimensional" space of them.

It is interesting to ask, can such a sequence be geometric, starting with 1? In other words, can you have such a sequence with

1,x,x^2,x^3,x^4, etc.?

A little thought shows that this happens exactly when x^2=x+1. This is a quadratic equation, and we can solve it to get the golden ratio and its conjugate. Note there are two independent solutions to the original problem.

Now linear algebra tells us that if you have two independent solutions to a 2-dimensional problem, every solution is a constant times one plus a constant times the other.

In particular, the sequence starting with 1,1 will be a constant times the first geometric sequence plus a constant times the second.
These constants are easy to solve for; you get 1/sqrt(5) and its negative.

But we have formulas for the geometric sequences! So we get a formula for the nth Fibonacci number as

[1/sqrt(5)] * [golden ratio]^n-[1/sqrt(5)] * [the conjugate]^n.

Answer 7

I'm assuming you know the actual numbers:

1,1,2,3,5,8,13,21, ...

The Golden Ratio is the ratio you get when you divide one fib # by the one before it. For example:

21/13 = 1.615

or

8/5 = 1.6

The trick is, these are only approximations, as you can try those, and get different ratios. The actual Golden Ratio is that when you imagine the Fibonacci Numbers going as far as you like, getting bigger and bigger. And that is where the formula comes in, which you don't seem to want, so I will stop there.

ps: divide your height by your bellybutton height. That is an example of the Golden Ratio. Again, only an approximation. My wife, daughters and I did that, and our ratio was a little higher than the Golden Ratio.

Answer 8

Fiboncacci sequence is very simply you take the previous numbers and add together to get the third.

Start with 1

0 + 1 = 1
so now your first two numbers are 1 and 1

1 + 1 = 2
sequence is now 1,1,2

adding the last 2 numbers will get you the next so:
1 + 2 = 3

1,1,2,3

2 + 3 = 5

3 + 5 = 8

Ect.

So the sequence goes:

1,1,2,3,5,8,13,21,34,55,89,144

The Golden Ration has to do with the same priciples only you are dividing instead of adding, I think.

Hope this helps

Answer 9

Fibonaci is a series of numbers where by the next term is the sum of the previous 2. As you iterate the sequence, f(n) / f(n-1) approaches the Golden Mean for higher and higher values of n (n is the number of iterations).

The most basic series is (0,1,1,2,3,5,8,13....) but you can start with any 2 positive numbers (one can be zero) and it will still approach the Golden Mean.

===Steven S=== below is wrong. The correct value for the Golden Mean is ( (sqrt(5) -1 ) / 2. This is the solution of the equation X^2 -x -1 = 0, which is .61803... The inverse is more commonly used which is 1.61803...

Interestingly, m-1 = 1/m or m-1/m = 1... Pretty unique!

Answer 10

34, 55 and 89. The next 3 are 144, 233 and 377. Each is formed by adding the previous 2 numbers in the sequence.