How do you find the equation for a non-linear pattern?

For example:

Let i=iteration number

Let n=the initial number used

Let x=the answer [when, of course, you sub in values for i and n])…

When i=0, x=n…

when i=1, x=(n+1)/n…

when i=2, x=(2n+1)/(n+1)…

when i=3, x=(3n+2)/(2n+1)…

when i=4, x=(5n+3)/(3n+2)…

when i=5, x=(8n+5)/(5n+3)…

basically (for the first iteration) find the reciprocal of the n value, then add one; then (for the second iteration) with your new number, find its reciprocal and add 1, etc. This is linked to the golden ratio and I think I might actually be in effect asking for the formula without phi.

Thanks

For example:

Let i=iteration number

Let n=the initial number used

Let x=the answer [when, of course, you sub in values for i and n])…

When i=0, x=n…

when i=1, x=(n+1)/n…

when i=2, x=(2n+1)/(n+1)…

when i=3, x=(3n+2)/(2n+1)…

when i=4, x=(5n+3)/(3n+2)…

when i=5, x=(8n+5)/(5n+3)…

basically (for the first iteration) find the reciprocal of the n value, then add one; then (for the second iteration) with your new number, find its reciprocal and add 1, etc. This is linked to the golden ratio and I think I might actually be in effect asking for the formula without phi.

Thanks

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