mathematics - Averaging numbers on the blackboard

Aatif sees the numbers $1 , 2 , 3 , .... , 2016$ written on the blackboard. In a move Aatif can pick any two numbers on the blackboard, erase them and write instead once their average. As an example, the numbers $1$ and $8$ may be replaced by $4 \frac{1}{2}$, and the numbers $2$ and $10$ may be replaced by $6$.

After $2015$ moves the blackboard only contains a single number. Can Aatif make his moves so that the final number is $2$?

Answer

Yes

First choose $2014$ and $2016$. Average = $2015$. Now take the $2015$s. Their average is $2015$.

Now choose $2015$ and $2013$. Average = $2014$.

Choose $2014$ and $2012$. Average = $2013$.

Note that we can keep on continuing this approach and end up with a situation like $1$, $2$ and $4$ in the end.

From here, choose $2$ and $4$. Average = $3$. Average of $1$ and $3$ is $2$.