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≥1T_{n+1} = 13 - T_n^2, \quad n \ge 1Tn+1=13−Tn2,n≥1
In 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.
Practise Edexcel A Level Maths Sequences and Series with exam-style questions for A Level Maths. 100 questions covering 3.1 Arithmetic Sequences, 3.2 Arithmetic Series, 3.3 Geometric Sequences, 3.4 Geometric Series, 3.5 Sum to Infinity, 3.6 Sigma Notation, 3.7 Recurrence Relations, and 3.8 Modelling with Series, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.