An industrial filtration system is used to remove heavy metals from wastewater. The mass of metals removed during the n n\,nth hour of operation is given by un u_n\,un mg, where u1,u2,u3,… u_1, u_2, u_3, \dots\,u1,u2,u3,… forms a geometric series. The maximum possible total mass of metals the system can ever remove is 192 mg.
In the second hour of operation, the system removes 45 mg.
The mass removed in the first hour, aaa, is greater than 100 mg.
Find the first term, aaa, and the common ratio, rrr, of the series.
Show that the mass removed in the n n\,nth hour can be written as
un=3n⋅523n−6 u_n = \frac{3^n \cdot 5}{2^{3n-6}} un=23n−63n⋅5Hence show that
log2un=n(log23−3)+(6+log25) \log_2 u_n = n(\log_2 3 - 3) + (6 + \log_2 5) log2un=n(log23−3)+(6+log25)