Question: Prove the following equations

(a) \(\sum\limits_{i=1}^{n}{\left( {{x}_{i}}-\bar{x} \right)=0}\)

(b) \(\sum\limits_{i=1}^{n}{\bar{x}=n\bar{x}}\)

(c) \(\sum\limits_{i=1}^{n}{{{\left( {{x}_{i}}-\bar{x} \right)}^{2}}\ne }\sum\limits_{i=1}^{n}{\left( {{x}_{i}}-\bar{x} \right)}\sum\limits_{i=1}^{n}{\left( {{x}_{i}}-\bar{x} \right)}\)

(d) \(\frac{\sum\limits_{i=1}^{n}{\left( {{x}_{i}}-\bar{x} \right)}\sum\limits_{n=1}^{n}{\left( {{y}_{i}}-\bar{y} \right)}}{\sum\limits_{i=1}^{n}{{{\left( {{x}_{i}}-\bar{x} \right)}^{2}}}}\ne \frac{\sum\limits_{i=1}^{n}{\left( {{y}_{i}}-\bar{y} \right)}}{\sum\limits_{i=1}^{n}{\left( {{x}_{i}}-\bar{x} \right)}}\)