Using your calculateor, look at the successive ratios (of the Fibonacci sequence) of one term to the next. Make a conjecture.

I don't exactly get what the questions asking. I tried listing the ratios, but I really didn't see any pattern. Thanks in advance. :)

I suspect the pattern you will see is that each term is the sum of the prior two terms.

check that.

1,1,2,3,5,8,13,21,... as bobpursley told you

as far as ratios go, try this

extend the list to some reasonable length
divide any number by its previous number.
The further out you go, the closer you will get to (1+√5)/2, which is called the Golden Ratio.

The Fibonacci sequence is a sequence of numbers where each number is the sum of the two preceding numbers. It starts with 0 and 1, so the Fibonacci sequence looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

To find the successive ratios, you divide each term by the previous term. For example, dividing 1 by 1 gives a ratio of 1, dividing 2 by 1 gives a ratio of 2, dividing 3 by 2 gives a ratio of 1.5, and so on.

If we continue calculating the ratios, we get the following results:

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538...
...

It might be hard to see a pattern right away, but if we keep calculating more ratios, we start to notice that the ratios are approaching a certain value. This value is known as the Golden Ratio, often represented by the Greek letter phi (Φ), which has an approximate value of 1.6180339887.

Based on the observed ratios, we can make the conjecture that as the Fibonacci sequence progresses, the successive ratios of one term to the next approach the value of the Golden Ratio, Φ.

To understand the pattern in the successive ratios of the Fibonacci sequence, let's first review what the Fibonacci sequence is. The Fibonacci sequence is a sequence of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1.

Here's the Fibonacci sequence for the first few terms:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

To calculate the successive ratios, we divide each term by the preceding one. For example, the ratio between the second and first term is 1/0 = undefined (since division by zero is not defined). The ratio between the third and second term is 1/1 = 1, the ratio between the fourth and third term is 2/1 = 2, and so on.

Let's write down the ratios for the first few terms of the Fibonacci sequence:

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 ≈ 1.6667
8/5 = 1.6
13/8 ≈ 1.625
21/13 ≈ 1.6154
34/21 ≈ 1.619
...

As you can see, these ratios don't have an obvious pattern. However, if we continue calculating more ratios, we will start to notice a pattern.

Let's calculate a few more ratios:

55/34 ≈ 1.6176
89/55 ≈ 1.6182
144/89 ≈ 1.6179
233/144 ≈ 1.6181
377/233 ≈ 1.618
...

As we calculate more ratios, we can observe that they are getting closer and closer to a specific value, approximately 1.618. This value is known as the golden ratio, often represented by the Greek letter φ (phi).

Therefore, based on the successive ratios of the Fibonacci sequence, the conjecture is that as we take larger and larger terms, the ratios will approach the golden ratio of approximately 1.618.