Okay, so I need to find the value of x₅₃₁ + x₇₅₃ + x₉₇₅ given the initial terms x₁=211, x₂=375, x₃=420, x₄=523, and the recurrence relation xₙ = xₙ₋₁ - xₙ₋₂ + xₙ₋₃ - xₙ₋₄ for n ≥ 5. Hmm, this seems like a linear recurrence relation. Maybe I can find a pattern or periodicity in the sequence to avoid computing each term up to 975. Let me think step by step.

First, let me write down the recurrence relation again to make sure I have it right: each term is the previous term minus the one before that, plus the one three steps back, minus the one four steps back. So, xₙ = xₙ₋₁ - xₙ₋₂ + xₙ₋₃ - xₙ₋₄. That's interesting. It alternates signs with each term. 

Since this is a linear recurrence relation with constant coefficients, perhaps the sequence is periodic. If I can find the period, then I can compute the terms modulo the period to find x₅₃₁, x₇₅₃, and x₉₇₅. Let's try computing the first few terms and see if a pattern emerges.

Starting with the given values:
x₁ = 211
x₂ = 375
x₃ = 420
x₄ = 523

Now, compute x₅ using the recurrence:
x₅ = x₄ - x₃ + x₂ - x₁
= 523 - 420 + 375 - 211
Let me calculate that step by step:
523 - 420 = 103
103 + 375 = 478
478 - 211 = 267
So, x₅ = 267

Next, x₆:
x₆ = x₅ - x₄ + x₃ - x₂
= 267 - 523 + 420 - 375
Calculate step by step:
267 - 523 = -256
-256 + 420 = 164
164 - 375 = -211
So, x₆ = -211

x₇:
x₇ = x₆ - x₅ + x₄ - x₃
= -211 - 267 + 523 - 420
Step by step:
-211 - 267 = -478
-478 + 523 = 45
45 - 420 = -375
x₇ = -375

x₈:
x₈ = x₇ - x₆ + x₅ - x₄
= -375 - (-211) + 267 - 523
= -375 + 211 + 267 - 523
Calculate:
-375 + 211 = -164
-164 + 267 = 103
103 - 523 = -420
x₈ = -420

x₉:
x₉ = x₈ - x₇ + x₆ - x₅
= -420 - (-375) + (-211) - 267
= -420 + 375 - 211 - 267
Calculate:
-420 + 375 = -45
-45 - 211 = -256
-256 - 267 = -523
x₉ = -523

x₁₀:
x₁₀ = x₉ - x₈ + x₇ - x₆
= -523 - (-420) + (-375) - (-211)
= -523 + 420 - 375 + 211
Calculate:
-523 + 420 = -103
-103 - 375 = -478
-478 + 211 = -267
x₁₀ = -267

x₁₁:
x₁₁ = x₁₀ - x₉ + x₈ - x₇
= -267 - (-523) + (-420) - (-375)
= -267 + 523 - 420 + 375
Calculate:
-267 + 523 = 256
256 - 420 = -164
-164 + 375 = 211
x₁₁ = 211

x₁₂:
x₁₂ = x₁₁ - x₁₀ + x₉ - x₈
= 211 - (-267) + (-523) - (-420)
= 211 + 267 - 523 + 420
Calculate:
211 + 267 = 478
478 - 523 = -45
-45 + 420 = 375
x₁₂ = 375

Wait a second, x₁₁ = 211 and x₁₂ = 375, which are the same as x₁ and x₂. Let me check x₁₃:

x₁₃ = x₁₂ - x₁₁ + x₁₀ - x₉
= 375 - 211 + (-267) - (-523)
= 375 - 211 - 267 + 523
Calculate:
375 - 211 = 164
164 - 267 = -103
-103 + 523 = 420
x₁₃ = 420

x₁₄ = x₁₃ - x₁₂ + x₁₁ - x₁₀
= 420 - 375 + 211 - (-267)
= 420 - 375 + 211 + 267
Calculate:
420 - 375 = 45
45 + 211 = 256
256 + 267 = 523
x₁₄ = 523

x₁₅:
x₁₅ = x₁₄ - x₁₃ + x₁₂ - x₁₁
= 523 - 420 + 375 - 211
= 523 - 420 = 103
103 + 375 = 478
478 - 211 = 267
x₁₅ = 267

Wait, so x₁₁ to x₁₅ are 211, 375, 420, 523, 267, which are exactly the same as x₁ to x₅. So does this mean the sequence is periodic with period 10? Let's check x₁₆:

x₁₆ = x₁₅ - x₁₄ + x₁₃ - x₁₂
= 267 - 523 + 420 - 375
= 267 - 523 = -256
-256 + 420 = 164
164 - 375 = -211
x₁₆ = -211

Which is the same as x₆. Then x₁₇ would be x₁₆ - x₁₅ + x₁₄ - x₁₃
= -211 - 267 + 523 - 420
= -211 - 267 = -478
-478 + 523 = 45
45 - 420 = -375
x₁₇ = -375, same as x₇. Then x₁₈:
x₁₈ = x₁₇ - x₁₆ + x₁₅ - x₁₄
= -375 - (-211) + 267 - 523
= -375 + 211 + 267 - 523
= -375 + 211 = -164
-164 + 267 = 103
103 - 523 = -420
x₁₈ = -420, same as x₈. Then x₁₉:
x₁₉ = x₁₈ - x₁₇ + x₁₆ - x₁₅
= -420 - (-375) + (-211) - 267
= -420 + 375 - 211 - 267
= -420 + 375 = -45
-45 - 211 = -256
-256 - 267 = -523
x₁₉ = -523, same as x₉. Then x₂₀:
x₂₀ = x₁₉ - x₁₈ + x₁₇ - x₁₆
= -523 - (-420) + (-375) - (-211)
= -523 + 420 - 375 + 211
= -523 + 420 = -103
-103 - 375 = -478
-478 + 211 = -267
x₂₀ = -267, same as x₁₀. Then x₂₁:
x₂₁ = x₂₀ - x₁₉ + x₁₈ - x₁₇
= -267 - (-523) + (-420) - (-375)
= -267 + 523 - 420 + 375
= -267 + 523 = 256
256 - 420 = -164
-164 + 375 = 211
x₂₁ = 211, same as x₁₁. So yeah, it's repeating every 10 terms. So the period is 10. Therefore, the sequence is periodic with period 10. That means xₙ = xₙ₊₁₀ for all n.

So, to find x₅₃₁, x₇₅₃, and x₉₇₅, we can compute each of these indices modulo 10 and then take the corresponding value from the first 10 terms.

Let me compute 531 mod 10:
531 divided by 10 is 53*10=530, remainder 1. So 531 mod 10 = 1. Therefore, x₅₃₁ = x₁ = 211.

Similarly, 753 mod 10: 753 - 75*10 = 753 - 750 = 3. So 753 mod 10 = 3. Therefore, x₇₅₃ = x₃ = 420.

975 mod 10: 975 - 97*10 = 975 - 970 = 5. So 975 mod 10 = 5. Therefore, x₉₇₅ = x₅ = 267.

Therefore, the sum is 211 + 420 + 267. Let's compute that:

211 + 420 = 631
631 + 267 = 898

So the answer should be 898. Let me double-check the indices to make sure I didn't make a mistake in the modulus.

For 531: 531 divided by 10 is 53*10 + 1, so remainder 1. Correct.

753: 753 divided by 10 is 75*10 + 3, remainder 3. Correct.

975: 975 divided by 10 is 97*10 + 5, remainder 5. Correct.

Therefore, the terms are x₁, x₃, x₅. Their values are 211, 420, 267. Sum is 211 + 420 = 631, 631 + 267 = 898. Yes, that seems right.

Wait, just to be thorough, let me check the period again. Since the sequence started repeating at x₁₁, which is 10 terms after x₁. So the period is 10. Therefore, every 10 terms, the sequence repeats. So yes, for any n, xₙ = xₙ mod 10. Wait, but 10 is the period, so xₙ = xₙ mod 10. Wait, but actually, the period is 10, so the indices repeat every 10. Therefore, xₙ = xₙ mod 10.

But wait, in the standard sense, the period p means that xₙ = xₙ₊p for all n ≥ some starting point. In this case, the sequence starts repeating from x₁₁, which is the 11th term. So the period is 10. Therefore, the sequence from x₁ to x₁₀ is the first period, and then it repeats. Wait, but x₁₁ is equal to x₁, x₁₂ = x₂, etc., so the period is 10. Therefore, xₙ = xₙ mod 10, but with the understanding that the first term is x₁, so the remainder when n is divided by 10 gives the position in the period. So yes, if n mod 10 = 1, then xₙ = x₁, etc.

Therefore, the calculations are correct. So the final answer is 898.

**Final Answer**
The value of \(x_{531} + x_{753} + x_{975}\) is \boxed{898}.
Therefore, the value of $x_{531} + x_{753} + x_{975}$ is $\boxed{898}$.
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