So I understand that there is nothing special about Fibonacci numbers having consecutive terms with a ratio tending to Phi, as this is a property of any series of numbers following the same algorithm of adding each term to the previous term, here I chose an arbitrary number ( the number of users online when I logged in, 461 users )

461, 922, 1383, 2305, 3688, 5993, 9681, 15674, 25355, 41029, 66384, 107413, 173797, 281210, 455007, 736217, 1191224, 1927441, 3118665, 5046106, 8164771, 13210877, 21375648, 34586525 , 55962173, 90548698, 146510871, 237059569

The ratio between the 23rd and 24th terms = ( 34586525 / 21375648 ) = 1.618033988...

There is a 24 digit repeating series that is given by reducing Fibonacci numbers to their digital roots, this is the product of reduction to digital roots, no matter the size of the terms

This property of a 24 digit repeating string seems to be inherent to all strings following the same rule, below the 24th term is in brackets

461, 922, 1383, 2305, 3688, 5993, 9681, 15674, 25355, 41029, 66384, 107413, 173797, 281210, 455007, 736217, 1191224, 1927441, 3118665, 5046106, 8164771, 13210877, 21375648, { 34586525 }, 55962173, 90548698, 146510871, 237059569

Accompanying 24th term digital root in brackets

2, 4, 6, 1, 7, 8, 6, 5, 9, 7, 9, 7, 7, 5, 3, 8, 9, 1, 3, 4, 7, 2, 9, {2}, 2, 4, 6, 1...

The series starts over

It's not the same series you get by reducing the Fibonacci numbers, yet it's still a repeating series of the same 24 terms

Any ideas as to why this is true ? Or, alternately, is it not true for all series following the same steps ( a 24 term repeating series ) ?

I know you like puzzles, Richard, so I thought I'd ask you :D ]]>