Let the sum of the first $$n$$ terms of a non-constant A.P., $$a_1, a_2, a_3, \ldots, a_n$$ be $$50n + \frac{n(n-7)}{2}A$$, where A is a constant. If $$d$$ is the common difference of this A.P., then the ordered pair $$(d, a_{50})$$ is equal to:

We are told that the sum of the first $$n$$ terms of a non-constant arithmetic progression (A.P.) is

$$S_n = 50n + \frac{n(n-7)}{2}\,A,$$

where $$A$$ is a constant. For any A.P. whose first term is $$a_1$$ and common difference is $$d$$, the standard formula for the sum of the first $$n$$ terms is

$$S_n = \frac{n}{2}\left[2a_1 + (n-1)d\right].$$

Equating the two expressions for $$S_n$$, we have

$$\frac{n}{2}\left[2a_1 + (n-1)d\right] = 50n + \frac{n(n-7)}{2}\,A.$$

Because the factor $$\dfrac{n}{2}$$ appears on both sides, we divide the entire equation by $$\dfrac{n}{2}$$ (noting that the A.P. is non-constant so $$n\neq0$$):

$$2a_1 + (n-1)d = 100 + A(n-7).$$

Expanding each side yields

$$2a_1 + nd - d = 100 + An - 7A.$$

Now we collect the terms involving $$n$$ and the constant terms separately. The left side contains $$nd$$ as the coefficient of $$n$$ and $$2a_1 - d$$ as the constant part, while the right side contains $$An$$ as the coefficient of $$n$$ and $$100 - 7A$$ as the constant part:

$$\underbrace{nd}_{\text{coefficient of }n} \;+\; \underbrace{(2a_1 - d)}_{\text{constant}} \;=\; \underbrace{An}_{\text{coefficient of }n} \;+\; \underbrace{(100 - 7A)}_{\text{constant}}.$$

Since this identity must hold for every value of $$n$$, the coefficients of corresponding powers of $$n$$ must match. Therefore, we obtain two separate equations:

Coefficient of $$n$$:  $$d = A,$$

Constant term:  $$2a_1 - d = 100 - 7A.$$

Substituting $$d = A$$ into the second equation, we find

$$2a_1 - A = 100 - 7A,$$

so

$$2a_1 = 100 - 7A + A = 100 - 6A,$$

and hence

$$a_1 = 50 - 3A.$$

To obtain the fiftieth term $$a_{50}$$, we recall the general term of an A.P.,

$$a_n = a_1 + (n-1)d.$$

With $$n = 50$$ and using $$d = A$$, we have

$$a_{50} = a_1 + 49d = (50 - 3A) + 49A = 50 + 46A.$$

Thus the ordered pair $$(d,\;a_{50})$$ is

$$(A,\,50 + 46A).$$

Hence, the correct answer is Option D.