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 $x_{531}+x_{753}+x_{975}$.
This problem involves a recursive sequence, which means that each term is defined in terms of previous terms. A common technique for solving problems with recursive sequences is to look for a pattern or a formula that describes the general term of the sequence. To do this, I will write out a few more terms of the sequence and see if I can notice any pattern. Using the given formula, I get \begin{align*}x_{5}&=x_{4}-x_{3}+x_{2}-x_{1}\\ &=523-420+375-211\\ &=267,\\ x_{6}&=x_{5}-x_{4}+x_{3}-x_{2}\\ &=267-523+420-375\\ =-211,\\ x_{7}&=x_{6}-x_{5}+x_{4}-x_{3}\\ &=-211-267+523-420\\ =-375,\\ x_{8}&=x_{7}-x_{6}+x_{5}-x_{4}\\ &=-375+211+267-523\\ =-420.\end{align*} I see that the terms are starting to repeat in reverse order. This suggests that the sequence is periodic, meaning that it repeats the same values after a certain number of terms. To confirm this, I will write out a few more terms and see if the pattern continues. I get \begin{align*}x_{9}&=x_{8}-x_{7}+x_{6}-x_{5}\\ &=-420+375-211+267\\ =11,\\ x_{10}&=x_{9}-x_{8}+x_{7}-x_{6}\\ &=11+420+375+211\\ =1017,\\ x_{11}&=x_{10}-x_{9}+x_{8}-x_{7}\\ &=1017-11-420+375\\ =961,\\ x_{12}&=x_{11}-x_{10}+x_{9}-x_{8}\\ &=961-1017+11+420\\ =375.\end{align*} I see that $x_{12}=x_{2}$, which confirms that the sequence is periodic with a period of 10. This means that every 10th term is the same as the corresponding term in the first 10 terms. For example, $x_{22}=x_{12}=x_{2}$, $x_{23}=x_{13}=x_{3}$, and so on. This makes the problem much easier, because I can use the modulo operation to find the corresponding term in the first 10 terms for any large value of $n$. The modulo operation, denoted by $\%$, gives the remainder when one number is divided by another. For example, $17\%5=2$, because $17=5\times3+2$. To find the corresponding term in the first 10 terms for any $n$, I can use the formula $x_{n}=x_{n\%10}$, where $n\%10$ is the remainder when $n$ is divided by 10. For example, $x_{37}=x_{37\%10}=x_{7}$, $x_{48}=x_{48\%10}=x_{8}$, and so on. Now, I can use this formula to find the values of $x_{531}$, $x_{753}$, and $x_{975}$. I get \begin{align*}x_{531}&=x_{531\%10}\\ &=x_{1}\\ &=211,\\ x_{753}&=x_{753\%10}\\ &=x_{3}\\ &=420,\\ x_{975}&=x_{975\%10}\\ &=x_{5}\\ &=267.\end{align*} Therefore, the value of $x_{531}+x_{753}+x_{975}$ is \begin{align*}x_{531}+x_{753}+x_{975}&=211+420+267\\ &=898.\end{align*}

