1-sqrt5 embedded in the Fibonacci numbers themselves through grouping?

Discussion in 'Number Theory' started by zvckry, Jan 12, 2025 at 8:52 PM.

  1. zvckry

    zvckry

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    I know the algebra formulas use the sqrt5+-1 to get phi so indirectly its related to the Fibonacci numbers, but can someone lead me to where it has been derived out of the numbers themselves in a consistent pattern without the algebra?

    Here is what i did

    n→∞lim fn+fn+1+fn+2/fn+3=1.23606

    46368 + 75025 + 121393= 242,786

    242,786/196418= 1.236067977476606
    another

    2178309+3524578+5702887=11,405,774

    11,405,774/9227465=1.236067977499779

    one more

    9425114757512643212875125+96151855463018422468774568+155576970220531065681649693
    =311,153,940,441,062,100,365,737,984

    311,153,940,441,062,100,365,737,984/251728825683549488150424261=1.2360679774997895732704694785035509073486321
     
    zvckry, Jan 12, 2025 at 8:52 PM
    #1
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