Chef's Quick Brain Teaser, revisited

[EternalOptimist]

What is the answer to questions 3 and 4?

mmmm, do they both result in a 1 ?

[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.

[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?