$\displaystyle \bar x_{n+1} = \frac{1}{n+1} \sum^{n+1}_{k=1} x_k$ ...and then, I'm stuck. No idea what to do next.
(a) $\displaystyle \bar x_{n} = \frac{1}{n} \sum^{n}_{k=1} x_k \Rightarrow \sum^{n}_{k=1} x_k = n \bar x_{n}$.
Therefore:
$\displaystyle \sum^{n+1}_{k=1} x_k = \left( \sum^{n}_{k=1} x_k \right) + x_{n+1} = n \bar x_{n} + x_{n+1}$.
Therefore $\displaystyle \bar x_{n+1} = \frac{n \bar x_{n} + x_{n+1}}{n+1}$.
You should have another go at (b) now ....