Let $$a$$, $$b$$ and $$c$$ be the 7$$^{th}$$, 11$$^{th}$$ and 13$$^{th}$$ terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $$\frac{a}{c}$$ is equal to:

Let us denote the first term of the arithmetic progression (A.P.) by $$A$$ and its common difference by $$d$$.

The $$n^{\text{th}}$$ term of an A.P. is given by the well-known formula

$$T_n \;=\; A + (n-1)d.$$

Using this, we translate the information given in the question:

We have

$$a \;=\; \text{7}^{\text{th}}\text{ term} \;=\; A + 6d,$$

$$b \;=\; \text{11}^{\text{th}}\text{ term} \;=\; A + 10d,$$

$$c \;=\; \text{13}^{\text{th}}\text{ term} \;=\; A + 12d.$$

Now the same three numbers $$a,\,b,\,c$$ also form three consecutive terms of a geometric progression (G.P.). In any G.P., for three consecutive terms, the middle term squared equals the product of the other two terms. Symbolically,

$$b^{2} \;=\; a\,c.$$

Substituting the expressions of $$a,\,b,\,c$$ from above, we get

$$\bigl(A + 10d\bigr)^{2} \;=\; \bigl(A + 6d\bigr)\,\bigl(A + 12d\bigr).$$

Expanding the left side completely,

$$\bigl(A + 10d\bigr)^{2} \;=\; A^{2} + 20Ad + 100d^{2}.$$

Next, expanding the right side,

$$\bigl(A + 6d\bigr)\,\bigl(A + 12d\bigr) \;=\; A^{2} + 12Ad + 6Ad + 72d^{2} \;=\; A^{2} + 18Ad + 72d^{2}.$$

Equating the two expansions, we have

$$A^{2} + 20Ad + 100d^{2} \;=\; A^{2} + 18Ad + 72d^{2}.$$

The $$A^{2}$$ terms cancel out from both sides, leaving

$$20Ad + 100d^{2} \;=\; 18Ad + 72d^{2}.$$

Bringing all terms to one side,

$$20Ad - 18Ad + 100d^{2} - 72d^{2} \;=\; 0,$$

$$2Ad + 28d^{2} \;=\; 0.$$

Now we can factor out $$2d$$ (remember the A.P. is non-constant, so $$d \neq 0$$), giving

$$2d\;\bigl(A + 14d\bigr) \;=\; 0.$$

Since $$2d \neq 0,$$ the remaining factor must vanish:

$$A + 14d \;=\; 0,$$

hence

$$A \;=\; -\,14d.$$

We now compute the required ratio $$\dfrac{a}{c}$$. Using $$A = -14d$$ we obtain

$$a \;=\; A + 6d \;=\; -14d + 6d \;=\; -\,8d,$$

$$c \;=\; A + 12d \;=\; -14d + 12d \;=\; -\,2d.$$

Therefore,

$$\frac{a}{c} \;=\; \frac{-8d}{-2d} \;=\; 4.$$

Hence, the correct answer is Option D.