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The problem provided involves a sequence defined by the recursive formula:

\[
a_n + a_{n+1} = (n - 1)^2
\]
with the given information that \( a_{10} = 50 \), and the task is to find the sum of the terms from the 8th to the 12th term of the sequence.

Let's proceed step by step to solve this:

### Step 1: Recursive formula setup

The recursive formula can be rewritten as:

\[
a_{n+1} = (n - 1)^2 - a_n
\]

This allows us to calculate subsequent terms using the value of \( a_{n} \).

### Step 2: Start from \( a_{10} = 50 \)

We know:
- \( a_{10} = 50 \)
- \( a_{9} + a_{10} = (9 - 1)^2 = 8^2 = 64 \)

Therefore:

\[
a_9 + 50 = 64 \quad \Rightarrow \quad a_9 = 64 - 50 = 14
\]

### Step 3: Calculate \( a_8 \)

Using the recursive formula again for \( a_8 \):

\[
a_8 + a_9 = (8 - 1)^2 = 7^2 = 49
\]

\[
a_8 + 14 = 49 \quad \Rightarrow \quad a_8 = 49 - 14 = 35
\]

### Step 4: Calculate \( a_{11} \)

For \( a_{11} \):

\[
a_{10} + a_{11} = (10 - 1)^2 = 9^2 = 81
\]

\[
50 + a_{11} = 81 \quad \Rightarrow \quad a_{11} = 81 - 50 = 31
\]

### Step 5: Calculate \( a_{12} \)

For \( a_{12} \):

\[
a_{11} + a_{12} = (11 - 1)^2 = 10^2 = 100
\]

\[
31 + a_{12} = 100 \quad \Rightarrow \quad a_{12} = 100 - 31 = 69
\]

### Step 6: Sum of terms from \( a_8 \) to \( a_{12} \)

Now we can sum the values from \( a_8 \) to \( a_{12} \):

\[
a_8 + a_9 + a_{10} + a_{11} + a_{12} = 35 + 14 + 50 + 31 + 69 = 199
\]

Thus, the correct answer is **199**.

---

Do you want further details or explanations on any step? Here are five related questions to explore:

1. How do you derive a recursive formula from a sequence?
2. Can the given recursive formula be solved explicitly for \( a_n \)?
3. What other types of sequences use recursive definitions?
4. How can we verify the correctness of recursive calculations?
5. What are the common methods to sum a sequence of terms?

**Tip:** When dealing with recursive sequences, always try to compute the terms step by step, working backwards if necessary from the provided initial condition.

Diketahui sebuah barisan yang memenuhi a_n + a_{n+1} = (n - 1)^2 dengan a_n adalah suku ke-n pada barisan dan a_{10} = 50. Jumlah suku ke-8 hingga suku ke-12 barisan tersebut adalah ....

Learn how to solve a sequence problem involving a recursive formula, where a_n + a_{n+1} = (n - 1)^2 and a_{10} = 50. This problem calculates the sum of the 8th to the 12th terms. Suitable for students in Grades 10-12.

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