I was just thinking about the golden ratio, and how it is a solution to the equation x^2 - x - 1 = 0.(adsbygoogle = window.adsbygoogle || []).push({});

So I wondered what it would be like to make a similar higher degree equation.

x^3 - x^2 - x - 1 = 0

To which the positive solution is 1.84.. is there any significance to this ratio?

I kept adding more terms.

x^4 - x^3 - x^2 - x - 1 = 0

x^5 - x^4 - x^3 - x^2 - x - 1 = 0

Etc.

These solutions, as I add more terms, grow more and more slowly and approach 2. I haven't found it to go past 2.

How can I show that the solution to the equation approaches 2 as the number of terms in the format I've shown grows without bound?

Another weird question from me, but I feel compelled to know the answers to these questions I have. :)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# How would you take the limit of this?

**Physics Forums | Science Articles, Homework Help, Discussion**