Key Concepts and Formulas

• Arithmetic Progression (A.P.): For terms $x, y, z$ in A.P., $2y = x + z$.
• Geometric Progression (G.P.): For terms $x, y, z$ in G.P., $y^2 = xz$.
• Logarithm Properties: $\log_a b = \frac{\log_x b}{\log_x a}$ (Change of base formula).
• Sum of an A.P.: $S_n = \frac{n}{2}[2a_1 + (n-1)d]$, where $a_1$ is the first term and $d$ is the common difference.

Step-by-Step Solution

Step 1: Use the A.P. condition for $a^3, b^3, c^3$. Given that $a^3, b^3, c^3$ are in A.P., by the definition of an A.P., we have: $2b^3 = a^3 + c^3 \quad (*)$

Step 2: Use the G.P. condition for the logarithms. Given that $\log_a b, \log_c a, \log_b c$ are in G.P., by the definition of a G.P., we have: $(\log_c a)^2 = (\log_a b)(\log_b c)$ Using the change of base formula for logarithms ($\log_x y = \frac{\log y}{\log x}$), we can rewrite the terms: $(\frac{\log a}{\log c})^2 = (\frac{\log b}{\log a})(\frac{\log c}{\log b})$ $(\frac{\log a}{\log c})^2 = \frac{\log c}{\log a}$ Multiplying both sides by $(\log c)^2 (\log a)$ to clear the denominators: $(\log a)^2 (\log a) = (\log c)^2 (\log c)$ $(\log a)^3 = (\log c)^3$ Taking the cube root of both sides gives: $\log a = \log c$ Since the logarithmic function is one-to-one, this implies: $a = c \quad (**)$

Step 3: Simplify the A.P. condition using $a=c$. Substitute $c=a$ into equation $(*)$: $2b^3 = a^3 + a^3$ $2b^3 = 2a^3$ $b^3 = a^3$ Taking the cube root of both sides: $b = a \quad (***)$ From equations $(**)$ and $(***)$, we conclude that $a = b = c$.

Step 4: Determine the first term and common difference of the second A.P. The first term of the second A.P. is given by $a_1 = \frac{a+4b+c}{3}$. Since $a=b=c$, we substitute $b=a$ and $c=a$: $a_1 = \frac{a+4a+a}{3} = \frac{6a}{3} = 2a$ The common difference of the second A.P. is given by $d = \frac{a-8b+c}{10}$. Substituting $b=a$ and $c=a$: $d = \frac{a-8a+a}{10} = \frac{-6a}{10} = -\frac{3a}{5}$

Step 5: Use the sum of the A.P. to find the value of $a$. The sum of the first 20 terms of this A.P. is given as $S_{20} = -444$. Using the formula $S_n = \frac{n}{2}[2a_1 + (n-1)d]$: $S_{20} = \frac{20}{2}[2(2a) + (20-1)(-\frac{3a}{5})] = -444$ $10[4a + 19(-\frac{3a}{5})] = -444$ $10[4a - \frac{57a}{5}] = -444$ To combine the terms inside the bracket, find a common denominator: $10[\frac{20a}{5} - \frac{57a}{5}] = -444$ $10[\frac{20a - 57a}{5}] = -444$ $10[\frac{-37a}{5}] = -444$ $2(-37a) = -444$ $-74a = -444$ Divide by -74 to solve for $a$: $a = \frac{-444}{-74} = 6$

Step 6: Calculate the value of $abc$. Since $a=b=c$ and we found $a=6$, we have $a=6, b=6, c=6$. Therefore, $abc = (6)(6)(6) = 6^3$. $abc = 216$

Common Mistakes & Tips

• Logarithm Properties: Ensure correct application of the change of base formula. Mistakes here can lead to incorrect relationships between $a$ and $c$.
• Algebraic Simplification: Be meticulous with algebraic manipulations, especially when dealing with fractions and exponents.
• A.P. and G.P. Definitions: Double-check the conditions for A.P. ($2y=x+z$) and G.P. ($y^2=xz$) to avoid errors.

Summary

We began by translating the given A.P. and G.P. conditions into algebraic equations. The G.P. condition for the logarithms simplified to $a=c$. Substituting this into the A.P. condition for the cubes led to $b=a$, thus establishing $a=b=c$. We then used the information about the second A.P. to express its first term and common difference in terms of $a$. Finally, we used the given sum of the first 20 terms of this second A.P. to solve for $a$, and subsequently calculated $abc$.

The final answer is \boxed{216}.