2015 NYJC P1 Q9
(a)
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 $n\in {{\mathbb{Z}}^{+}}$.
(i) If the product of ${{x}_{5}}$ and ${{x}_{21}}$ is $4096$, find the value of $\sum\limits_{k=1}^{25}{{{\log }_{4}}{{x}_{k}}}$.
[4]
(ii) Show that ${{y}_{1}}$, ${{y}_{2}}$, ${{y}_{3}}$, … is an arithmetic sequence.
[2]
(b)
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]
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