A specialized industrial centrifuge extracts a mineral from a solution such that the mass extracted each hour forms a geometric series. The theoretical maximum mass that can be extracted over an infinite time period is 150 grams. In the first hour, the centrifuge extracts aaa grams, where a>80a > 80a>80. In the second hour, it extracts 36 grams.
Determine the value of the first term, aaa, and the common ratio, rrr, of the series.
Show that the mass extracted in the nnnth hour, MnM_nMn, can be written as
Mn=2n⋅325n−2 M_n = \frac{2^n \cdot 3^2}{5^{n-2}} Mn=5n−22n⋅32Hence show that
log2Mn=n(1−log25)+(2log23+2log25) \log_2 M_n = n(1 - \log_2 5) + (2 \log_2 3 + 2 \log_2 5) log2Mn=n(1−log25)+(2log23+2log25)
