I'm a bit stumped at the moment.
I'm trying to generate a sequence where the first number is 1.9, the tenth is 3, and the 20th is between $10^{5-6}$ . There should be a function $f(x_1) = 1.9$, $f(x_{10}) = 3$, and $f(x_{20}) >10000$. I've been thinking an exponential function, but maybe not. Ideally it'd be a function that doesn't have any discontinuities. The $x's$ should be equally spaced.
So: $x_1^n = 1.9$, $x_{10}^n = 3$, $x_{10}-x_1 = d$, $x_{20}>10^5$, and $x_{20}<10^6$
Intuitively, there should be a whole set of $d$'s and $n$'s that would work.
I'm guessing that $d$ and $n$ would be uniquely identified if I specified $x_{20}$, but I can't show this at the moment.
Perhaps the solution is obvious and I'm just momentarily blind to it. Help appreciated.
Edit: I should mention that, when I simulate, I end up needing exponents $n$ that are larger than machine precision if I stick with an exponential function and specify $x_{20} = 10000^{1/n}$. So an exponential seems not be the way to go.
My guess is you want something like $x(n)=a+be^{cn}$ and solve for the coefficients after substituting your desired values.
Seeing your condition for the values to be equally spaced, then what you ask cannot be done. Equally spaced means $x(n)=a+bx$, and you have three restrictions and only two parameters.