2015 NYJC P1 Q9

A geometric sequence \({{x}_{1}}\), \({{x}_{2}}\), \({{x}_{3}}\), … has first term \(a\) and common ratio \(r\), where \(a>0\), \(r>0\). The sequence of numbers \({{y}_{1}}\), \({{y}_{2}}\), \({{y}_{3}}\), … satisfies the relation \({{y}_{n}}={{\log }_{4}}\left( {{x}_{n}} \right)\) for \(n\in {{\mathbb{Z}}^{+}}\).

The output of an oil field in any year is \(6\%\) less than in the preceding year. Prove that the total output of the oil mined from the oil field cannot exceed \(17\) times the output in the first year.[2]

It is decided to close the mine when the total output exceeds \(16\) times the output in the first year. Suppose the oil field was first mined in the year 1997, determine the earliest year in which the oil field will be closed.[2]