Something to do with Fibonnaci sequence.?

Question

A) As u all know fibonnaci sq is 1,1,2,3,5,8,13,21....................

I have been wondering what is the end of this sequence, i checked the internet and it gave me a number which is irrational number. (root5-1)/2 or (root5+1)/2, forgot
Anyone here explain this?


B)

If 1/3 = 0.33333 rational

Multyiply 3 on both sides

3/3 = 0.9999 rational

1 = 0.999999 rational???

Any one here explain this?

Answer 1

1) First, you are talking about the *ratio* of consecutive terms, not the last term. The series can extend indefinitely, but the ratio between terms approaches the golden ratio of phi = (1 + sqrt(5))/2

2) First you need to be sure to talk about 0.33333... infinitely repeating, not just 0.33333.

Then definitely 0.33333... = 1/3

And 0.99999.... = 1

This is just another way to represent 1.

Answer 2

A)

Neither (root5-1)/2 nor (root5+1)/2 are NOT the end of the Fibonacci sequence (because the sequence is infinite).

I think this is what you were thinking is the nth term of the Fibonacci sequence:
(1/root5)[((1 + root5)/2)^n - ((1 - root5)/2)^n]

If you plug in any integer n, you will get that nth Fibonacci number within the sequence.

Also, the value (1+root5)/2, which is denoted by the Greek symbol phi (φ), is the golden ratio. Phi has a special connection to the Fibonacci sequence: As the sequence grows infinitely large, the ratio between a Fibonacci number and the term before it approaches phi (φ), which is once again (1+root5)/2 or 1.618033988749894848...

B)

What you just did was prove a mathematical fact. But what you're failing to mention is that the 0.3333 and the 0.9999 are INFINITELY repeating decimals

since 1/3 = 0.3333333333333333 (3 repeating)
1 = 0.9999999999999999 (9 repeating)

This is a mathematical FACT that has been proven many different ways; people just mistakingly reject it.

Answer 3

A) There is no real "end" to the sequence. Generally speaking, the Fibonacci sequence is recursive and uses two previous numbers in the sequence and takes their sum to get the next number in the sequence.

Therefore the Fibonacci sequence goes like this:

Let a1= 1
Let a2=1
a3= a1 + a2 = 1+1 = 2

1, 1, 2, 3, 5, 8, 13, 21, ... , (a(n-1) + a(n)). Where a(n-1) is the a-1 th term of the sequence and a(n) is the nth term of the sequence)

If you look at the ratios of the terms in the sequence they approach what is known as the golden ratio, which are the numbers you found.

B)

The question 1=0.999999999.... is equivalent. First, let x= 0.999999999.....

Multiply both sides of the equation by 10.

10x = 9.99999999999......

Subtract the two equations......

10x = 9.99999999......
- x =-0.99999999......
----------------------------

9x= 9

Divide both sides by 9.

x = 1

Therefore 1 = 0.999999.....

Additionally, any decimal that can be put into a fraction is a rational number.

Answer 4

A) The equation for phi is the second one, (root5+1)/2. The golden ratio (phi), is defined as the ratio that results when a line is divided so that the whole line has the same ratio to the larger segment as the larger segment has to the smaller segment. Expressed mathematically, normalizing the larger part to unit length, it is the positive solution of the equation:

B) 1/3 is actually irrational because the number of 3s in 0.33... is infinite. As you add more 3s to the end of the decimal place, you increase the accuracy of the approximation. Computer programs and calculators have to take this approximation into account and round off a number like 0.9999 to 1.0

Answer 5

A) The Fibonacci sequence continues on forever. Why? Because each number is the result of the previous two numbers:
1=0+1
2=1+1
3=2+1
5=3+2
8=5+3
13=8+5
... and so on
0,1,1,2,3,5,8,13,21...

B) A way to understand this is if you look at your calculator, and add 1/3 to 1/3. You now have 0.6666666666666666.... which your calculator represents as 0.6666666666667 (because it has rounded up the last number (because it can't represent the repeating decimal). Add 0.3333333333333 to that, and it is clear that the sum is 1.

Answer 6

The Fibonacci sequence is infinite. The number you found on the internet is the ratio towards which two consecutive numbers of the sequence tends.

A rational number is a number which can be written in the form p/q, where p & q are integers, and q is a positive integer.

Since 1 = (3x1)/3 = 3/3, one is a rational number, and 0.999999(etc) is also a rational number.

Answer 7

I don't know about A ..
but about B, 1/3 = 0.33333333333333333333333333333.....until infinity , but since everyone wants to use the fraction value in calculation, we need to make it digit limited, that's why they take smaller approximate value

Answer 8

Its jsut the way it is, but liek if it was draw on a chart of somehting, it wouldnt be noticed...cause we are talking about like

0.000...1

Like its a VERY small number so it doesnt really matter