I’m generating the following sequence , :
$$y_i = 1/i$$
where :
$$i = 1 \dots \infty$$
Given that I have the calculated-value at point x :
$$y_x$$
how should I calculate the next value $$y_{x+1}$$
I don’t know the index ‘x‘, I know only the calculated value.
For example, if $$y_x = 0.167$$ then $$y_{x+1} = 0.143$$ i.e. 1/6 implies 1/7
After a calculation I keep only the calculated value, not the current index.
Here is the formula :
$$ y_{x+n} = \frac {1} {{\frac 1 {y_{x}}}+n}$$
Detailed solution :
$$y_x = k; k= {\frac 1x}; x = {\frac 1k}$$
$$ y_{x+n} = \frac {1} {x+n} = \frac {1} {{\frac 1k}+n} = \frac {1} {{\frac 1 {y_{x}}}+n}$$
