How the mean responds to changes in the data

A set of \(n\) numbers has mean \(m\), so their total is \(nm\).

1. Show that adding \(5\) to every number gives a new mean of \(m+5\).[2]
2. State the new mean if every number is doubled.[1]
3. An extra number \(v\) is included. Show that the new mean is \(m+\dfrac{v-m}{n+1}\), and hence explain why the mean increases exactly when \(v>m\).[3]
4. Twelve numbers have mean \(15\). Find the number that must be included to make the mean \(16\).[1]