The depth of water in a tidal sensor tank, dnd_ndn mm, measured at the start of each hour nnn, is modeled by the recurrence relation
dn+1=P−Qdn d_{n+1} = P - Q d_n dn+1=P−Qdnwhere PPP and QQQ are constants and d1=6d_1 = 6d1=6.
Find expressions in terms of PPP and QQQ for (i) d2d_2d2 (ii) d3d_3d3
Given that ∑n=13dn=60\sum_{n=1}^{3} d_n = 60∑n=13dn=60 and P=5Q+8P = 5Q + 8P=5Q+8
Show that Q2−4Q−38=0Q^2 - 4Q - 38 = 0Q2−4Q−38=0.
Hence find the smaller possible value of d2d_2d2, giving your answer in the form a−ba - \sqrt{b}a−b where aaa and bbb are integers.
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.