<Question>
Given that \begin{align*}x_{1}&=211,\\ x_{2}&=375,\\ x_{3}&=420,\\ x_{4}&=523,\ \text{and}\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{align*}find the value of $x_{531}+x_{753}+x_{975}$.
</Question>

<think>
<Plan_1>
Okay, so I have this problem where I need to find the value of x₅₃₁ + x₇₅₃ + x₉₇₅. The sequence is defined with the first four terms given: x₁ = 211, x₂ = 375, x₃ = 420, x₄ = 523. Then, for n ≥ 5, each term is defined recursively as xₙ = xₙ₋₁ - xₙ₋₂ + xₙ₋₃ - xₙ₋₄. Hmm, that seems a bit complicated, but maybe there's a pattern or cycle in the sequence that repeats after a certain number of terms. If I can figure out the cycle length, then I can find the values of x₅₃₁, x₇₅₃, and x₉₇₅ by determining their positions within the cycle. Let me try to compute the first few terms and see if a pattern emerges.

First, let me write down the initial terms:

<prompt_1.1> Write down the initial terms of the sequence as given in the problem. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
x₁ = 211

x₂ = 375

x₃ = 420

x₄ = 523

Now, let's compute x₅ using the recursive formula:

x₅ = x₄ - x₃ + x₂ - x₁

Plugging in the values:

x₅ = 523 - 420 + 375 - 211

Let me calculate that step by step:

523 - 420 = 103

103 + 375 = 478

478 - 211 = 267

So, x₅ = 267.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, x₆:

<prompt_2.1> Calculate the next few terms of the sequence using the recursive formula and look for a pattern. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
x₆ = x₅ - x₄ + x₃ - x₂

= 267 - 523 + 420 - 375

Calculating step by step:

267 - 523 = -256

-256 + 420 = 164

164 - 375 = -211

Hmm, x₆ = -211. Interesting, that's the negative of x₁.

Moving on to x₇:

x₇ = x₆ - x₅ + x₄ - x₃

= -211 - 267 + 523 - 420

Calculating:

-211 - 267 = -478

-478 + 523 = 45

45 - 420 = -375

So, x₇ = -375. That's the negative of x₂.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Let's compute x₈:

<prompt_3.1> Continue calculating the sequence and look for a pattern. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
x₈ = x₇ - x₆ + x₅ - x₄

= -375 - (-211) + 267 - 523

Calculating:

-375 + 211 = -164

-164 + 267 = 103

103 - 523 = -420

x₈ = -420. That's the negative of x₃.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
x₉:

<prompt_4.1> Identify the pattern in the sequence and confirm if the cycle holds. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
x₉ = x₈ - x₇ + x₆ - x₅

= -420 - (-375) + (-211) - 267

Calculating:

-420 + 375 = -45

-45 - 211 = -256

-256 - 267 = -523

x₉ = -523. That's the negative of x₄.

x₁₀:

x₁₀ = x₉ - x₈ + x₇ - x₆

= -523 - (-420) + (-375) - (-211)

Calculating:

-523 + 420 = -103

-103 - 375 = -478

-478 + 211 = -267

x₁₀ = -267. Which is the same as x₅.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
So, x₁₀ = -267, which was x₅. Let's check x₁₁:

<prompt_5.1> Verify if the cycle continues and determine the cycle length. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
x₁₁ = x₁₀ - x₉ + x₈ - x₇

= -267 - (-523) + (-420) - (-375)

Calculating:

-267 + 523 = 256

256 - 420 = -164

-164 + 375 = 211

x₁₁ = 211. Which is the same as x₁.
</execution_5.1>
</Execution_5>

<Plan_6>
Based on execution_5.1:
So, from x₁₀ onwards, the sequence repeats x₅, x₆, x₇, x₈, x₉, etc. So, the cycle seems to be 6 terms long. Let's confirm this by computing a few more terms.

<prompt_6.1> Confirm the cycle by computing more terms and finalize the cycle length. </prompt_6.1>
</Plan_6>

<Execution_6>
<execution_6.1>
x₁₁ = 211 (which is x₁)

x₁₂ = x₁₁ - x₁₀ + x₉ - x₈

= 211 - (-267) + (-523) - (-420)

Calculating:

211 + 267 = 478

478 - 523 = -45

-45 + 420 = 375

x₁₂ = 375 (same as x₂)

x₁₃ = x₁₂ - x₁₁ + x₁₀ - x₉

= 375 - 211 + (-267) - (-523)

Calculating:

375 - 211 = 164

164 - 267 = -103

-103 + 523 = 420

x₁₃ = 420 (same as x₃)

x₁₄ = x₁₃ - x₁₂ + x₁₁ - x₁₀

= 420 - 375 + 211 - (-267)

Calculating:

420 - 375 = 45

45 + 211 = 256

256 + 267 = 523

x₁₄ = 523 (same as x₄)

x₁₅ = x₁₄ - x₁₃ + x₁₂ - x₁₁

= 523 - 420 + 375 - 211

Calculating:

523 - 420 = 103

103 + 375 = 478

478 - 211 = 267

x₁₅ = 267 (same as x₅)

So, yes, the sequence repeats every 6 terms. The cycle is: 211, 375, 420, 523, 267, -211, -375, -420, -523, -267, -211, etc. Wait, hold on. Wait, when we look at x₁ to x₁₄, the terms from x₁ to x₄ are 211, 375, 420, 523. Then x₅ is 267, x₆ is -211, x₇ is -375, x₈ is -420, x₉ is -523, x₁₀ is -267, x₁₁ is 211, x₁₂ is 375, x₁₃ is 420, x₁₄ is 523. So actually, the cycle is 211, 375, 420, 523, 267, -211, -375, -420, -523, -267, and then back to 211? Wait, but x₁₁ is 211, which is x₁, x₁₂ is x₂, etc. So the cycle length is 10? Wait, let's list the terms:

n : xₙ

1 : 211

2 : 375

3 : 420

4 : 523

5 : 267

6 : -211

7 : -375

8 : -420

9 : -523

10 : -267

11 : 211

12 : 375

13 : 420

14 : 523

15 : 267

16 : -211

... So, from n=1 to n=10, the terms are 211, 375, 420, 523, 267, -211, -375, -420, -523, -267. Then n=11 to n=14 repeat 211, 375, 420, 523, etc. So the cycle length is 10.
</execution_6.1>
</Execution_6>

<Plan_7>
Based on execution_6.1:
Wait, but when we look at x₁ to x₁₀, then x₁₁ is x₁ again, so the cycle is 10 terms. Let me check again:

<prompt_7.1> Re-evaluate the cycle length by listing terms and confirm the cycle. </prompt_7.1>
</Plan_7>

<Execution_7>
<execution_7.1>
Cycle:

1: 211

2: 375

3: 420

4: 523

5: 267

6: -211

7: -375

8: -420

9: -523

10: -267

11: 211 (same as 1)

12: 375 (same as 2)

... So yes, the cycle is 10 terms. Therefore, the sequence is periodic with period 10.

So, the key takeaway here is that the sequence repeats every 10 terms. Therefore, to find xₙ for any n, we can compute n modulo 10 (or 10, adjust for zero-based indexing) and find the equivalent term in the first cycle.

But let's verify this with the terms we calculated.

x₁=211, x₁₁=211, x₂₁ should also be 211, right? Let's check if that's the case.

But since we only calculated up to x₁₅, maybe we can compute a few more terms to ensure the cycle continues.

x₁₆ = x₁₅ - x₁₄ + x₁₃ - x₁₂

= 267 - 523 + 420 - 375

Calculating:

267 - 523 = -256

-256 + 420 = 164

164 - 375 = -211

x₁₆ = -211 (same as x₆)

x₁₇ = x₁₆ - x₁₅ + x₁₄ - x₁₃

= -211 - 267 + 523 - 420

Calculating:

-211 - 267 = -478

-478 + 523 = 45

45 - 420 = -375

x₁₇ = -375 (same as x₇)

x₁₈ = x₁₇ - x₁₆ + x₁₅ - x₁₄

= -375 - (-211) + 267 - 523

Calculating:

-375 + 211 = -164

-164 + 267 = 103

103 - 523 = -420

x₁₈ = -420 (same as x₈)

x₁₉ = x₁₈ - x₁₇ + x₁₆ - x₁₅

= -420 - (-375) + (-211) - 267

Calculating:

-420 + 375 = -45

-45 - 211 = -256

-256 - 267 = -523

x₁₉ = -523 (same as x₉)

x₂₀ = x₁₉ - x₁₈ + x₁₇ - x₁₆

= -523 - (-420) + (-375) - (-211)

Calculating:

-523 + 420 = -103

-103 - 375 = -478

-478 + 211 = -267

x₂₀ = -267 (same as x₁₀)

x₂₁ = x₂₀ - x₁₉ + x₁₈ - x₁₇

= -267 - (-523) + (-420) - (-375)

Calculating:

-267 + 523 = 256

256 - 420 = -164

-164 + 375 = 211

x₂₁ = 211 (same as x₁)

Yep, so the cycle is indeed 10 terms. Therefore, the sequence is periodic with period 10.
</execution_7.1>
</Execution_7>

<Plan_8>
Based on execution_7.1:
Now, to find x₅₃₁, x₇₅₃, and x₉₇₅, we can compute each index modulo 10 to find their positions in the cycle.

First, let's handle x₅₃₁:

<prompt_8.1> Calculate x₅₃₁ using modulo operation. </prompt_8.1>

Now, x₇₅₃:

<prompt_8.2> Calculate x₇₅₃ using modulo operation. </prompt_8.2>

Next, x₉₇₅:

<prompt_8.3> Calculate x₉₇₅ using modulo operation. </prompt_8.3>
</Plan_8>

<Execution_8>
<execution_8.1>
531 divided by 10. Let's compute 531 ÷ 10. 10*53 = 530, so 531 = 10*53 + 1. Therefore, 531 mod 10 is 1. So, x₅₃₁ corresponds to the first term in the cycle, which is x₁ = 211.
</execution_8.1>

<execution_8.2>
531 divided by 10. Let's compute 531 ÷ 10. 10*53 = 530, so 531 = 10*53 + 1. Therefore, 531 mod 10 is 1. Therefore, x₅₃₁ corresponds to the first term in the cycle, which is 211.

Similarly, x₇₅₃:

753 divided by 10. 10*75 = 750, so 753 - 750 = 3. Therefore, 753 mod 10 is 3. So, the third term in the cycle, which is 420.

Wait, hold on. Wait, the cycle is 10 terms, starting from x₁. So, if the remainder is 1, it's x₁; remainder 2, x₂; ... ; remainder 0 (if any), which would correspond to 10. But in our case, 531 mod 10 is 1, so x₁. 753 mod 10 is 3, so x₃. 975 mod 10 is 5 (since 975 divided by 10 is 97*10 +5). So, x₅.

Therefore, x₅₃₁ = x₁ = 211

x₇₅₃ = x₃ = 420

x₉₇₅ = x₅ = 267

Therefore, the sum x₅₃₁ + x₇₅₃ + x₉₇₅ = 211 + 420 + 267. Let's compute that:

211 + 420 = 631

631 + 267 = 898

So, the answer would be 898.
</execution_8.2>

<execution_8.3>
So, 975 divided by 10: 975 /10 = 97*10 +5, so remainder 5. Therefore, x₉₇₅ is the 5th term in the cycle, which is 267.

Therefore, x₅₃₁ + x₇₅₃ + x₉₇₅ = (-523) + (-375) + 267. Let's compute that:

First, -523 - 375 = -900 - 2 (Wait, 523 + 375 is 898, so -523 -375 = -898). Then, -898 + 267 = -631.
</execution_8.3>
</Execution_8>

<Plan_9>
Based on execution_8.1, execution_8.2, execution_8.3:
Wait, but let me double-check my calculations to make sure I didn't make any mistakes.

First, confirm the cycle length is 10.

<prompt_9.1> Verify the cycle length and recalculate the sum to ensure accuracy. </prompt_9.1>

But let me check once again if the cycle is indeed 10. Let's list the terms again:

<prompt_9.2> Re-verify the cycle length by listing terms and confirm the sum. </prompt_9.2>

But wait, let me check if the cycle is correctly identified. When we computed up to x₁₅, x₁₆, etc. , the terms were repeating every 10 terms.
So, for example, x₁₁ = x₁, x₁₂ = x₂, so the cycle is indeed 10. Therefore, the indices modulo 10 give the position in the cycle.

But let me confirm the modulo operation for each term:

<prompt_9.3> Confirm the modulo operation for each term and verify the sum. </prompt_9.3>

But let me check the arithmetic again:

<prompt_9.4> Verify the arithmetic calculations for the sum of x₅₃₁, x₇₅₃, and x₉₇₅. </prompt_9.4>

But just to be thorough, let me recompute x₇₅₃ and x₉₇₅ using the cycle:

<prompt_9.5> Recompute x₇₅₃ and x₉₇₅ using the cycle and confirm the sum. </prompt_9.5>

But just to ensure there's no mistake in the cycle length, let's compute a few more terms beyond x₂₀ to see if the pattern holds.

<prompt_9.6> Compute additional terms to ensure the cycle pattern holds and confirm the sum. </prompt_9.6>

Therefore, the answer is 898.

Wait, but let me check once more if the problem statement is correctly interpreted.

<prompt_9.7> Finalize the answer and ensure the problem statement is correctly interpreted. </prompt_9.7>
</Plan_9>

<Execution_9>
<execution_9.1>
Yes, from x₁ to x₁₀, then x₁₁ repeats x₁, so the cycle is 10. So, every term with n ≡ k mod 10 is equal to xₖ. Therefore:

x₅₃₁: 531 mod 10 = 1 → x₁ = 211

x₇₅₃: 753 mod 10 = 3 → x₃ = 420

x₉₇₅: 975 mod 10 = 5 → x₅ = 267

So, adding them: 211 + 420 + 267. 211 + 420 is 631, plus 267 is 898.
</execution_9.1>

<execution_9.2>
n : xₙ

1 : 211

2 : 375

3 : 420

4 : 523

5 : 267

6 : -211

7 : -375

8 : -420

9 : -523

10: -267

11: 211

12: 375

13: 420

14: 523

15: 267

16: -211

17: -375

18: -420

19: -523

20: -267

21: 211

... So, yes, every 10 terms, it repeats. So, the cycle is 10 terms. Therefore, the calculations using modulo 10 are correct.

Therefore, x₅₃₁ is 211, x₇₅₃ is 420, x₉₇₅ is 267. Sum is 211 + 420 + 267 = 898.
</execution_9.2>

<execution_9.3>
x₅₃₁: 531 divided by 10 gives remainder 1. Correct, since 531 = 53*10 + 1. So, remainder 1. Therefore, x₁ = 211.

x₇₅₃: 753 divided by 10 gives remainder 3. 753 = 75*10 + 3. So, remainder 3. Therefore, x₃ = 420.

x₉₇₅: 975 divided by 10 gives remainder 5. 975 = 97*10 +5. So, remainder 5. Therefore, x₅ = 267.

So, adding them up: 211 + 420 + 267.

211 + 420: 200 + 400 = 600, 11 + 20 = 31. So, 600 + 31 = 631. Then 631 + 267: 600 + 200 = 800, 31 + 67 = 98. 800 + 98 = 898. So, total is 898.
</execution_9.3>

<execution_9.4>
x₅₃₁: 531 divided by 10 gives remainder 1. Correct.

x₇₅₃: 753 divided by 10 gives remainder 3. Correct.

x₉₇₅: 975 divided by 10 gives remainder 5. Correct.

So, the terms are 211, 420, 267. Sum is 211 + 420 = 631; 631 + 267 = 898.
</execution_9.4>

<execution_9.5>
x₅₃₁: 531 mod 10 = 1 → x₁ = 211

x₇₅₃: 753 mod 10 = 3 → x₃ = 420

x₉₇₅: 975 mod 10 = 5 → x₅ = 267

Sum: 211 + 420 + 267 = 898. That's correct.

But hold on, x₉₇₅: 975 divided by 10 is 97*10 +5, so remainder 5. So, x₅ is 267. Correct.

But let me check if the problem asks for x₅₃₁ + x₇₅₃ + x₉₇₅. Yes, that's correct. So, 211 + 420 + 267 = 898.
</execution_9.5>

<execution_9.6>
x₂₁ = x₂₀ - x₁₉ + x₁₈ - x₁₇

= -267 - (-523) + (-420) - (-375)

Calculating:

-267 + 523 = 256

256 - 420 = -164

-164 + 375 = 211

x₂₁ = 211 (same as x₁)

x₂₂ = x₂₁ - x₂₀ + x₁₉ - x₁₈

= 211 - (-267) + (-523) - (-420)

Calculating:

211 + 267 = 478

478 - 523 = -45

-45 + 420 = 375

x₂₂ = 375 (same as x₂)

So, yes, the cycle is indeed 10 terms. Therefore, the earlier calculations hold.

Therefore, x₅₃₁ = 211, x₇₅₃ = 420, x₉₇₅ = 267. Sum is 211 + 420 + 267 = 898.

But wait, just to ensure, maybe compute x₉₇₅ again. 975 mod 10 is 5, so x₅ is 267.

Yes. Therefore, the final answer is 898.
</execution_9.6>

<execution_9.7>
The problem defines x₁ to x₄ as given, and for n ≥5, xₙ = xₙ₋₁ - xₙ₋₂ + xₙ₋₃ - xₙ₋₄. Then, compute x₅₃₁ + x₇₅₃ + x₉₇₅. So, yes, the terms are correctly identified.

But let me check once more the cycle positions:

For x₅₃₁: 531 divided by 10 gives remainder 1, so first term, 211.

x₇₅₃: 753 divided by 10 gives remainder 3, third term, 420.

x₉₇₅: 975 divided by 10 gives remainder 5, fifth term, 267.

Sum: 211 + 420 + 267. 211 + 420 is 631, plus 267 is 898. Correct.

But let me check if the problem is in Chinese or if there was a translation issue. Wait, the user provided the problem in English, so no issue. The terms are correctly calculated. The cycle is 10. The modulo operations are correct. Therefore, the answer should be 898.
</execution_9.7>
</Execution_9>

<Plan_10>
Based on execution_9.7:
**Final Answer**
\boxed{898}
</Plan_10>

<Execution_10>
No prompt to execute.
</Execution_10>

<Final_answer>
Therefore, the final answer is $\boxed{898}$.
</Final_answer>
</think>