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.