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)308 exam-style questions on OCR (MEI) A Level Maths 1.6 Sequences and Series, covering 1.6.1 Binomial expansion for positive integer n, 1.6.2 Factorial and combinations notation, 1.6.3 Binomial expansion for rational n (A-level only), 1.6.4 Binomial expansion of (a + bx)^n via factoring (A-level only), 1.6.5 Binomial approximating polynomials (A-level only), 1.6.6 What a sequence is; finite and infinite (A-level only), 1.6.7 Generate a sequence by formula or recurrence (A-level only), 1.6.8 A series as a sum of terms (A-level only), 1.6.9 Sigma notation (A-level only), 1.6.10 Increasing, decreasing and periodic sequences (A-level only), 1.6.11 Convergent and divergent sequences (A-level only), 1.6.12 Arithmetic sequences and series (A-level only), 1.6.13 Standard formulae for arithmetic series (A-level only), 1.6.14 Geometric sequences and series (A-level only), 1.6.15 Standard formulae for geometric series (A-level only), 1.6.16 Sum to infinity of a geometric series (A-level only), and 1.6.17 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.