Let's break this problem into steps.

The task is to evaluate the summation $\sum_{i=0}^4 X_{2i}$ using the given data, and round the result to 1 significant figure.

Step 1: Identify $X_{2i}$

We need to compute the values of $X_{2i}$ for $i = 0, 1, 2, 3, 4$. The subscript $2i$ refers to even indices in the data set.

• When $i = 0$, $2i = 0$. However, there is no $X_0$ in the table, so it is likely excluded.
• When $i = 1$, $2i = 2$. From the table, $X_2 = -1.2$.
• When $i = 2$, $2i = 4$. From the table, $X_4 = \frac{1}{5} = 0.2$.
• When $i = 3$, $2i = 6$. From the table, $X_6 = 12$.
• When $i = 4$, $2i = 8$. From the table, $X_8 = -0.1 \cdot \sum_{i=0}^3 i$.

Step 2: Evaluate $\sum_{i=0}^3 i$ for $X_8$

The summation $\sum_{i=0}^3 i = 0 + 1 + 2 + 3 = 6$.

Thus, $X_8 = -0.1 \cdot 6 = -0.6$.

Step 3: Summation $\sum_{i=0}^4 X_{2i}$

Now, compute the summation of the $X_{2i}$ values: $\sum_{i=0}^4 X_{2i} = X_2 + X_4 + X_6 + X_8 = -1.2 + 0.2 + 12 - 0.6$

Simplify step by step: $-1.2 + 0.2 = -1.0$ $-1.0 + 12 = 11.0$ $11.0 - 0.6 = 10.4$

Step 4: Round to 1 Significant Figure

Rounding $10.4$ to 1 significant figure gives 10.

Final Answer:

$\boxed{10}$

Would you like further clarification on any step or related questions? Here are some questions for deeper understanding:

1. How are indices and subscripts used in summation problems like this one?
2. What is the significance of rounding in numerical problems?
3. How can you simplify summations involving arithmetic series like $\sum_{i=0}^3 i$?
4. How would the result change if one term were incorrect or missing in the table?
5. How do you verify calculations involving multiple steps?

Tip: Always double-check values extracted from a table or data set to avoid errors in subsequent calculations.