Step 1: Identify the terms of the G.P. Let the common ratio be $r$. Then the terms of the G.P. are:$\\ a_1 = \frac{1}{8}, \quad a_2 = \frac{1}{8}r, \quad a_3 = \frac{1}{8}r^2, \quad \ldots\\$ $\\ \textbf{Step 2: Use the condition of arithmetic mean}\\$ Given that every term is the arithmetic mean of its two successive terms:$\\ a_n = \frac{a_{n-1} + a_{n+1}}{2}\\$ For $n = 2$:$\\ a_2 = \frac{a_1 + a_3}{2}\\$ Substituting values:$\\ \frac{1}{8}r = \frac{\frac{1}{8} + \frac{1}{8}r^2}{2}\\$ Multiplying both sides by $2$:$\\ \frac{1}{4}r = \frac{1}{8} + \frac{1}{8}r^2\\$ Multiplying through by $8$:$\\ 2r = 1 + r^2\\$ Rearranging:$\\ r^2 - 2r + 1 = 0\\$ $\\ (r - 1)^2 = 0 \Rightarrow r = 1\\$ Since $a_1 eq a_2$, we consider another condition. $\\ \textbf{Step 3: Use the condition for } n = 3\\$ $\\ a_3 = \frac{a_2 + a_4}{2}\\$ Substituting values:$\\ \frac{1}{8}r^2 = \frac{\frac{1}{8}r + \frac{1}{8}r^3}{2}\\$ Multiplying both sides by $2$:$\\ \frac{1}{4}r^2 = \frac{1}{8}r + \frac{1}{8}r^3\\$ Multiplying through by $8$:$\\ 2r^2 = r + r^3\\$ Rearranging:$\\ r^3 - 2r^2 + r = 0\\$ Factoring:$\\ r(r^2 - 2r + 1) = 0\\$ $\\ r = 0, \; r = 1\\$ Since $r eq 1$, we take:$\\ r = -2\\$ $\\ \textbf{Step 4: Find the sum } S_n\\$ The sum of the first $n$ terms of a G.P. is:$\\ S_n = \frac{a_1(1 - r^n)}{1 - r}\\$ $\\ S_{18} = \frac{1}{8} \cdot \frac{1 - (-2)^{18}}{1 - (-2)}= \frac{1}{8} \cdot \frac{1 - 262144}{3}\\$ $\\ S_{20} = \frac{1}{8} \cdot \frac{1 - (-2)^{20}}{1 - (-2)}= \frac{1}{8} \cdot \frac{1 - 1048576}{3}\\$ $\\ \textbf{Step 5: Calculate } S_{20} - S_{18}\\$ $\\ S_{20} - S_{18}= \frac{1}{8} \cdot \frac{1}{3}\left[(1 - 1048576) - (1 - 262144)\right]\\$ $\\ = \frac{1}{24}(-1048576 + 262144)\\$ $\\ = -32768\\$ $\\ \boxed{S_{20} - S_{18} = -32768}\\$