How to find a rule from a table

Question

X | Y
1 | 18
2 | 24
3 | 42
4 | 96

How does one find a rule from a table like this? The only way I am able to find rules to ones like this is by using guess and check but I know there must be a better way.

Answer 1

$f(1)=18$

$f(2)=24=18+6$

$f(3)=42=24+18=24+3.6=18+6+3.6=18+(3+1).6$

$f(4)=96=42+54=42+3^2.6=18+(3+1).6+3^2.6=18+(3^2+3+1).6$

$...$

$\displaystyle f(n)=18+6.\sum_{i=0}^{n-2}3^i=18+6.\frac{(3^{(n-1)}-1)}{2}$

is a good guess.

And I think it is reasonable, if you want a function on $\mathbb{R}$, to say:

$\displaystyle f(x)=18+6.\frac{(3^{(x-1)}-1)}{2}$

As for the part

"The only way I am able to find rules to ones like this is by using guess and check but I know there must be a better way."

There are well-known "canonical" ways to extrapolate a function (by polynomials, for example). But my guess is that, in your case, what I did is most appropriate, since the rule is quite evident.