Chef's Quick Brain Teaser, revisited

[BrollyBonce]

1. Chef asked what was next in the series 1, 2, 6, 42, 1806, ? The answer appeared later in that thread.

2. conned_tractor asked where the series 1, 2, 1.5, 1.666, 1.6, ... converges. There was some controversy about the question, but the answer was a special number known as the Golden Ratio.

Those questions are actually related and I thought EO or sasguru or NickFitz or someone would spot the relationship and comment on it. They didn't, so I gave a clue in the form of a question:

3. "In the context of the first question in this thread, what is the significance of the number 0.6180339887498949......"

That should have prompted someone to say, "Hey, not only is that number rhubarb regarding question 1, it is also rhubarb regarding question 2!"

4. When you have answered question 3 (and not before), why is that number significant to question 2 and, therefore, how are questions 1 and 2 related?

Come on you clever people. What is the answer to questions 3 and 4?

[OwlHoot]

1. Chef asked what was next in the series 1, 2, 6, 42, 1806, ? The answer appeared later in that thread.

2. conned_tractor asked where the series 1, 2, 1.5, 1.666, 1.6, ... converges. There was some controversy about the question, but the answer was a special number known as the Golden Ratio.

Those questions are actually related and I thought EO or sasguru or NickFitz or someone would spot the relationship and comment on it. They didn't, so I gave a clue in the form of a question:

3. "In the context of the first question in this thread, what is the significance of the number 0.6180339887498949......"

That should have prompted someone to say, "Hey, not only is that number rhubarb regarding question 1, it is also rhubarb regarding question 2!"

4. When you have answered question 3 (and not before), why is that number significant to question 2 and, therefore, how are questions 1 and 2 related?

Come on you clever people. What is the answer to questions 3 and 4?

0.618033... is (SQRT(5) - 1)/2, which is the Golden Ratio less 1.

As for the other questions, a sequence like this has more than one right answer, although most of us probably assumed the pattern Chef had in mind was 1, 1.2 = 2, 2.3 = 6, 6.7 = 42, ...

If you look up the sequence at the Sloane's On-Line encyclopedia of integer sequences, you'll find several solutions, and not much mention of Fibonacci numbers or the Golden Ratio.

[EternalOptimist]

What is the answer to questions 3 and 4?

mmmm, do they both result in a 1 ?

[conned tractor]

3. "In the context of the first question in this thread, what is the significance of the number 0.6180339887498949......"

That should have prompted someone to say, "Hey, not only is that number rhubarb regarding question 1, it is also rhubarb regarding question 2!"

4. When you have answered question 3 (and not before), why is that number significant to question 2 and, therefore, how are questions 1 and 2 related?

Come on you clever people. What is the answer to questions 3 and 4?

I still can't see the link in 1). Am looking forward to seeing the answer. When will you enlighten us if nobody gets it?

And WTF is "rhubarb" - Is it a replacement for brown sticky stuff?

[BrilloPad]

Is there a prize for the right answer?

[Churchill]

Is there a prize for the right answer?

Oh goody, I like prizes!

[BrollyBonce]

0.618033... is the Golden Ratio less 1.

Well done. That is the answer to the first part of question 4.

What is its significance to the number series in Question 1?

Edit:
Do not get hung up on either the first number series or this number being related to the Golden Ratio. You first need to identify the very simple relationship between 0.6180339887498949... and the first number series - and Google is not the solution, merely observation and curiosity.

[BrollyBonce]

I still can't see the link in 1).

That's because you are looking too hard.

Am looking forward to seeing the answer.

I had assumed you had asked your question because you were already aware of the link between the two.

When will you enlighten us if nobody gets it?

Once sasguru and NF have had a go, and/or I can bear the waiting no longer.

And WTF is "rhubarb" - Is it a replacement for brown sticky stuff?

Just a substitute for 'some words'.

[Churchill]

Well done. That is the answer to the first part of question 4.

What is its significance to the number series in Question 1?

Edit:
Do not get hung up on either the first number series or this number being related to the Golden Ratio. You first need to identify the very simple relationship between 0.6180339887498949... and the first number series - and Google is not the solution, merely observation and curiosity.

All I can see in question 1 is

x = x * (x + 1);

[EternalOptimist]

All I can see in question 1 is

x = x * (x + 1);

I thought it was

x = (x * x) + x