3 A tree at the bottom of a garden needs to be reduced in height. The tree is known to increase in height by 15 centimetres each year. On the first day of every year, the height is measured and the tree is immediately trimmed by \(3 \%\) of this height. When the tree is measured, before trimming on the first day of year 1 , the height is 6 metres.
Let \(H _ { n }\) be the height of the tree immediately before trimming on the first day of year \(n\).

1. Explain, in the context of the problem, why the height of the tree may be modelled by the recurrence relation $$H _ { n + 1 } = 0.97 H _ { n } + 0.15 , \quad H _ { 1 } = 6 , \quad n \in \mathbb { Z } ^ { + }$$
2. Prove by induction that \(H _ { n } = 0.97 ^ { n - 1 } + 5 , \quad n \geqslant 1\)
3. Explain what will happen to the height of the tree immediately before trimming in the long term.
4. By what fixed percentage should the tree be trimmed each year if the height of the tree immediately before trimming is to be 4 metres in the long term?