The equation $f(x)= \frac{3^x+1}{2}$ for all positive integers x

The equation $f(x)= \frac{3^x+1}{2}$ for all positive integers x

The equation $f(x)= \frac{3^x+1}{2}$ for all positive integers $x$
generates a sequence such that the difference of every consecutive f(x)
forms another say g(x) such that $g(x) = 3^{x-1}$. What I did is to find
$f'(x)$ and expecting that it is equal to $3^x$, but it did not.
For example: if $x= 1,2,3,4,5$, $f(x) = 2,5,14,41,122$. I observed that $
5-2 =3; 14-5 = 3^2;41-14=3^3 ; 122-41= 3^4 $ Is this true for all? thanks