The first few terms:
1
5
5/sqrt(27)
25/sqrt(727)
125/sqrt(54679)
3125/sqrt(85285383)
The numerators are 5^F(n-1) where F(n) is the nth fibonacci number.
This can be shown by taking a(i-1)=A/sqrt(x), a(i)=B/sqrt(y), the next term
a(i+1)=AB/sqrt(yA^2+xB^2+xy)
Multiplication makes the powers add and the first two terms have numerators 5^0 and 5^1 the powers are fibonacci.
Since fibonacci numbers have an explicit form, the numerators are taken care of.
The denominators grow fast and I have not yet deduced a pattern. The first few terms from a spreadsheet:
Edit: you can't paste a google sheet into perplexus. You can do one cell at a time:
1
5
0.9622504486
0.9271986769
0.5345640287
0.3383862594
0.152864801
0.04849236174
0.007319250282
0.0003545016858
0.000002594616903
0.0000000009197960082
0
0
0
Edited on December 15, 2023, 3:13 pm
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Posted by Jer
on 2023-12-15 15:09:25 |
Partial solution
