Non-linear pattern equations

[Tuna4]

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

 

[stapel]

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

The method used will likely vary with the information provided and the tools available. I doubt there is a "one size fits all" algorithm.

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.

From the steps you've posted, I think the rule is actually take the previous numerator and convert it to the new denominator; then add the previous numerator and denominator together to get the new numerator. The "understood" first denominator is "1". (Just adding a "1" to the numerator (?) does not lead to the displayed results.)

This is linked to the golden ratio and I think I might actually be in effect asking for the formula without phi.

\(\displaystyle \phi\,\) is just the shorthand for the radical expression. If you don't want to use the variable shorthand, use the radical expression instead.

Or, if I'm misunderstanding what you're trying to do, kindly please reply with the full and exact text of the exercise, the complete instructions, and a clear listing of your efforts so far. Thank you!

 

[Tuna4]

Sorry for my lack of attention to detail

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

What I meant was to add the value of the denominator to the numerator.

 

[Ishuda]

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

Experience and, to paraphrase stapel, the experience you have had will sometimes tend to lead you in different directions.

For this particular pattern, I would go look up the Fibonacci Sequence
https://en.wikipedia.org/wiki/Fibonacci_number
and see if it lead anywhere since it appears to me that
\(\displaystyle x_i\, =\, \frac{n\, F_{i+1}\, +\, F_i}{n\, F_i\, +\, F_{i-1}};\,\, i\, \ge\, 1\)
where F_i is the Fibonacci Sequence (with F₀=0)