Let the coefficients of the middle terms in the expansion of $$\left(\frac{1}{\sqrt{6}} + \beta x\right)^4$$, $$(1 - 3\beta x)^2$$ and $$\left(1 - \frac{\beta}{2}x\right)^6$$, $$\beta > 0$$ respectively form the first three terms of an A.P. If $$d$$ is the common difference of this A.P., then $$50 - \frac{2d}{\beta^2}$$ is equal to _____

We need to find the middle term coefficients of three expansions and use them to form an A.P.

For $$\left(\dfrac{1}{\sqrt{6}} + \beta x\right)^4$$ (even power 4), the middle term is the 3rd term ($$r = 2$$):

$$\binom{4}{2}\left(\dfrac{1}{\sqrt{6}}\right)^2 (\beta x)^2 = 6 \cdot \dfrac{1}{6} \cdot \beta^2 x^2 = \beta^2 x^2$$

So the coefficient $$a_1 = \beta^2$$.

For $$(1 - 3\beta x)^2$$ (even power 2), the middle term is the 2nd term ($$r = 1$$):

$$\binom{2}{1}(1)(-3\beta x) = -6\beta x$$

So the coefficient $$a_2 = -6\beta$$.

For $$\left(1 - \dfrac{\beta}{2}x\right)^6$$ (even power 6), the middle term is the 4th term ($$r = 3$$):

$$\binom{6}{3}\left(-\dfrac{\beta}{2}\right)^3 x^3 = 20 \cdot \left(-\dfrac{\beta^3}{8}\right) x^3 = -\dfrac{5\beta^3}{2} x^3$$

So the coefficient $$a_3 = -\dfrac{5\beta^3}{2}$$.

Since $$a_1, a_2, a_3$$ form an A.P., we have $$2a_2 = a_1 + a_3$$:

$$2(-6\beta) = \beta^2 + \left(-\dfrac{5\beta^3}{2}\right)$$

$$-12\beta = \beta^2 - \dfrac{5\beta^3}{2}$$

Since $$\beta > 0$$, we can divide both sides by $$\beta$$:

$$-12 = \beta - \dfrac{5\beta^2}{2}$$

$$\dfrac{5\beta^2}{2} - \beta - 12 = 0$$

$$5\beta^2 - 2\beta - 24 = 0$$

Using the quadratic formula: $$\beta = \dfrac{2 \pm \sqrt{4 + 480}}{10} = \dfrac{2 \pm 22}{10}$$

Since $$\beta > 0$$, we get $$\beta = \dfrac{24}{10} = \dfrac{12}{5}$$.

Now we find the common difference $$d = a_2 - a_1 = -6\beta - \beta^2 = -6 \cdot \dfrac{12}{5} - \dfrac{144}{25} = -\dfrac{72}{5} - \dfrac{144}{25} = -\dfrac{360 + 144}{25} = -\dfrac{504}{25}$$.

Finally, $$50 - \dfrac{2d}{\beta^2} = 50 - \dfrac{2 \times \left(-\dfrac{504}{25}\right)}{\dfrac{144}{25}} = 50 - \dfrac{-\dfrac{1008}{25}}{\dfrac{144}{25}} = 50 + \dfrac{1008}{144} = 50 + 7 = 57$$.

Hence, the answer is $$57$$.