Let's solve this step by step.

Problem Breakdown

1. The terms $3a_1$, $2a_2$, and $a_3$ form an arithmetic sequence (AP).
   • For an arithmetic sequence, the difference between consecutive terms is constant: $2a_2 - 3a_1 = a_3 - 2a_2$
2. $(S_n)$ is the sum of the first $n$ terms of a geometric sequence with ratio $r \neq 1$.
3. We need to determine:
   • The value of $r$,
   • The value of $\frac{S_4}{S_2}$.

Step 1: Relation from the Arithmetic Sequence

The common difference condition gives: $2a_2 - 3a_1 = a_3 - 2a_2$ Simplify: 4a_2 = 3a_1 + a_3 \tag{1}

For a geometric sequence: $a_2 = a_1 r \quad \text{and} \quad a_3 = a_1 r^2$ Substitute $a_2$ and $a_3$ into Equation (1): $4(a_1 r) = 3a_1 + a_1 r^2$ Divide through by $a_1$ (since $a_1 \neq 0$): $4r = 3 + r^2$ Rearrange into standard quadratic form: $r^2 - 4r + 3 = 0$ Factorize: $(r - 3)(r - 1) = 0$ Thus: $r = 3 \quad \text{or} \quad r = 1$ Since $r \neq 1$, we have: $r = 3$

Step 2: Partial Sum of a Geometric Sequence

The $n$-th partial sum of a geometric sequence is given by: $S_n = a_1 \frac{1 - r^n}{1 - r} \quad \text{for} \quad r \neq 1$

For $S_2$:

$S_2 = a_1 \frac{1 - r^2}{1 - r}$

Substitute $r = 3$: $S_2 = a_1 \frac{1 - 3^2}{1 - 3} = a_1 \frac{1 - 9}{-2} = a_1 \frac{-8}{-2} = 4a_1$

For $S_4$:

$S_4 = a_1 \frac{1 - r^4}{1 - r}$

Substitute $r = 3$: $S_4 = a_1 \frac{1 - 3^4}{1 - 3} = a_1 \frac{1 - 81}{-2} = a_1 \frac{-80}{-2} = 40a_1$

Ratio $\frac{S_4}{S_2}$:

$\frac{S_4}{S_2} = \frac{40a_1}{4a_1} = 10$

Final Answer:

$r = 3 \quad \text{and} \quad \frac{S_4}{S_2} = 10$

Do you want a deeper explanation of any of these steps? Here are some follow-up questions you might consider:

1. How is the quadratic equation derived in this problem?
2. Why is $r \neq 1$ a necessary condition for solving the sum?
3. How does the partial sum formula for a geometric sequence work?
4. What changes in the solution if $r = -1$?
5. Could this problem have other solutions under different constraints?

Tip: Always check the common difference carefully when working with arithmetic sequences derived from other types of sequences!