does the quotient of the adjacent numbers of the fibonacci sequence always give the no. 1.618?

Question

in the book the 'da vinci code' it is said so.But the first two numbers of the sequece 1-1-2-3-5-8-13-21 doesn't give the no. 1.618 when divided. Y is it so?

Answer 1

fiction books are typically fiction.

Answer 2

The number 1.618 is a decimal approximation for what is known as the Golden Ratio. The Golden Ratio is really the number (1 + sqrt(5))/2 = 1.618033988749894. . .

Look at the ratio of the first few consecutive Fibonacci numbers

2/1 = 2.00000
3/2 = 1.50000
5/3 = 1.66666
8/5 = 1.60000
13/8 = 1.62500
21/13 = 1.61538
34/21 = 1.61905
55/34 = 1.61765

No matter how far you go into the list, there will never be two consecutive Fibonacci numbers that will give you the Golden Ratio exactly. But, the longer you go into the Fibonacci sequence, the closer you can get to the Golden Ratio.

89/55 = 1.61818
144/89 = 1.61798
233/144 = 1.61806
377/233 = 1.61803

I had to go out to the 13th and 14th consecutive Fibonacci numbers (233 and 377) before the ratio of the Fibonacci numbers agrees with the first 5 digits of the Golden Ratio.

Hope this helps!

Answer 3

Here is simple way to prove the golden ratio. Since we know that for the fibonacci sequence, the ratio becomes closer and closer to a constant, and we can assume the constant is r.
By the definition of the fibonacci sequence,
f(n) = f(n-1)+f(n-2)
Divide both sides by f(n-1)
f(n)/f(n-1) = 1 + 1/[f(n-1)/f(n-2)]......(1)
As n->∞, we have
f(n)/f(n-1) = f(n-1)/f(n-2) = r......(2)
Plug in (2) for (1),
r = 1 + 1/r
Multiply both sides by r, and collect all the terms in one side,
r^2-r-1 = 0
Solve for r with quadratic formula,
r = (1+ √5)/2, the so-called golden ratio.

Answer 4

As you go further out in the Fibonacci sequence, the ratio of one term to the previous one approaches a limit, which is so famous it has a name: phi
The exact formula for phi is:
[sqrt(5)+1]/2 = approximately 1.618033989...
Phi is also called the "Golden Mean", the proportion the Greeks felt made the most beautiful rectangles (like the Parthenon, its sides are in that proportion). This is just one of its many properties.
The Golden Mean appears all over mathematics, feel free to Google it to learn more!
Have fun.....

Answer 5

The quotient will approach the so-called golden ratio as the terms of the sequence increase. If you calculate the firs few you'll see that they begin to gravitate to this number. e.g., 5/3 = 1.6667; 8/5 = 1.6, 13/8 = 1.625, and 21/13 = 1.615.

Read an intrductory article on the 'golden ratio' for more.

Answer 6

No. But as you go further and further out in the sequence the value approaches 1.618.

Answer 7

The book is a work of fiction. So everything in the book is fake.