Just in case a picture helps...
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
For part (a):
You're given:
$\displaystyle \overline{X}=n^{-1}\sum_{i=1}^{n}X_i$ (expression (1)) and you need to prove that: $\displaystyle \sum_{i=1}^{n}(X_i-\overline{X})=0$ (expression (2))
Substitute (1) in (2):
$\displaystyle \sum_{i=1}^{n} \left(X_i-n^{-1}\sum_{i=1}^{n}X_i\right)$
To see clearly what happens we write down a few terms of the summation:
$\displaystyle [X_1-n^{-1}(X_1+X_2+\ldots+X_n)]+[X_2-n^{-1}(X_1+X_2+\ldots+X_n)]+\ldots+[X_n-n^{-1}(X_1+X_2+\ldots+X_n)]$
this is equal to:
$\displaystyle (X_1+X_2+\ldots+X_n)-nn^{-1}(X_1+X_2+\ldots+X_n)=0$