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)308 exam-style questions on OCR A Level Maths 1.4 Sequences and Series, covering 1.4.1 Binomial expansion for positive integer n, 1.4.2 Link to binomial probabilities, 1.4.3 Binomial expansion for rational n (A-level only), 1.4.4 Validity of the expansion (A-level only), 1.4.5 Sequences (A-level only), 1.4.6 Increasing, decreasing and periodic sequences (A-level only), 1.4.7 Sigma notation (A-level only), 1.4.8 Arithmetic sequences and series (A-level only), 1.4.9 Geometric sequences and series (A-level only), 1.4.10 Sum to infinity of a geometric series (A-level only), and 1.4.11 Modelling with sequences and series (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.