We are given that the first term $u_1 = 1$, and the common ratio is $r$. Additionally, we have two pieces of information about the partial sums of the geometric sequence:

• $S_2 = 20$ (the sum of the first two terms),
• $S_4 = 200$ (the sum of the first four terms).

Step 1: Express the sums

The sum of the first $n$ terms of a geometric sequence is given by the formula: $S_n = u_1 \frac{1 - r^n}{1 - r}$ where $u_1$ is the first term, and $r$ is the common ratio.

Since $u_1 = 1$, the formula simplifies to: $S_n = \frac{1 - r^n}{1 - r}$

Now, let's use this formula for the given sums.

Step 2: Use $S_2 = 20$

For $S_2 = 20$: $S_2 = \frac{1 - r^2}{1 - r} = 20$

Step 3: Use $S_4 = 200$

For $S_4 = 200$: $S_4 = \frac{1 - r^4}{1 - r} = 200$

Step 4: Solve the system of equations

We now have two equations:

1. $\frac{1 - r^2}{1 - r} = 20$
2. $\frac{1 - r^4}{1 - r} = 200$

We can solve this system of equations to find the value of $r$. Let’s simplify and solve this system.It seems there were no solutions directly found from the two equations. This suggests that there might be specific constraints on the value of $r$ or an alternative method to solve this system could be better.

Let's instead approach the problem by isolating the variable $r$ from each equation and look for relationships between them. Starting from the first equation: $\frac{1 - r^2}{1 - r} = 20$ Multiplying both sides by $1 - r$, we get: $1 - r^2 = 20(1 - r)$ Expanding and simplifying this will give us a quadratic equation. I'll solve it first and then move to the second equation to check the consistency.From the first equation, we have two possible solutions for the common ratio $r$: $r = 1 \quad \text{or} \quad r = 19$

Step 5: Check consistency with $S_4 = 200$

We will now substitute these values of $r$ into the second equation to check which one satisfies the condition $S_4 = 200$.

I'll test both $r = 1$ and $r = 19$ in the second equation.It turns out that neither $r = 1$ nor $r = 19$ satisfies the second equation $S_4 = 200$. This suggests that there might be an issue with the setup or interpretation of the problem.

We could approach this with a different method, such as directly solving for the partial sums using numerical techniques. Would you like to proceed with that, or would you like more clarification on the current approach?

In the meantime, here are some questions you might consider:

1. Would rechecking the initial conditions clarify the sums?
2. Could there be a different formula setup that better fits this sequence?
3. What other constraints could affect the value of $r$?
4. Can the sum of the terms approach a different pattern?
5. How might a numerical method help here?

Tip: Always recheck for the correctness of formulas and simplify each step to avoid misinterpretations.