Consider measurements taken from $g$ groups, with the mean $\mu_g$ being the mean of the measurements for group $g$.
I am trying to determine the number of possible ways that these means could be compared (in statistics these are denoted as contrasts). The potential comparisons could be the difference between two means ($\mu_1$ vs. $\mu_2$), the unweighted average of a set of means and a single other mean ( $\frac{\mu_1 + \mu_2}{2}$ vs. $\mu_3$) or the unweighted average of two sets of means ( $\frac{\mu_1 + \mu_2}{2}$ vs. $\frac{\mu_3 + \mu_4}{2}$).
The problem asks to prove that the number of possible such comparisons is $$ K_g = \frac{1}{2}(3^g - 2^{g+1} + 1) $$ with the hint given that
$$ K_g = 3K_{g - 1} + 2^{g - 1} - 1 $$ I can show that the above two relationships hold between $K_g$ using a proof by induction, but I am yet to prove that $K_g$ does accurately represent the number of possible comparisons.
