The temperature TnT_nTn (in ∘C{^\circ}\text{C}∘C) inside a specialized climate-controlled incubator on day n n\,n is modeled by the recurrence relation:
Tn+1=13−Tn2,n≥1 T_{n+1} = 13 - T_n^2, \quad n \ge 1 Tn+1=13−Tn2,n≥1In the case where the initial temperature is set to T1=4T_1 = 4T1=4:
(i) Calculate the value of T3T_3T3.
(ii) Deduce the value of T100T_{100}T100.
State a different value for T1T_1T1, other than T1=4T_1 = 4T1=4, which would result in the same value for T100T_{100}T100 as found in part (a)(ii). Justify your answer.
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.