The Square Root of Five and Fibonacci Series

I was messing with the Fibonacci Series and discovered a link to the square root of five. I took the first four numbers of the sequence, starting with 1,2,3,5. I added the first and last together(5 and 1) and the middle two together (2 and 3) and then divide the first by the second (6/5). I then put it into decimal form (1.2).

I kept doing this with every four numbers in the sequence. The numbers kept converging towards 1.23606. I finally reached 1.23606 at the quadruplet set 144, 233, 377, 610.

The square root of five is 2.23606. I don't know if this means anything, but I thought it was neat anyway, especially since phi is related to the square root of five in some ways,and phi is related to the Fibonacci Series.

phi= square root of 5 +1/2

There are other formulas that relate phi to the square root of five as well.

I also think it's revealing that the first quadruplet number turned out to be 6/5, which is another mysterious number in the hidden structure of the Universe. I will do a post on 6/5 later.