The temperature of a specialized metal alloy at the end of stage nnn of a cooling process is denoted by TnT_nTn, measured in Kelvin. The process is modeled by the recurrence relation
Tn+1=ωTn+84 T_{n+1} = \omega T_n + 84 Tn+1=ωTn+84where T1=1200T_1 = 1200T1=1200 and ω\omegaω is a constant.
Find an expression, in terms of ω\omegaω, for T2T_2T2.
It is given that the temperature at the end of the third stage is T3=822T_3 = 822T3=822.
Show that ω\omegaω satisfies the equation
200ω2+14ω−123=0 200\omega^2 + 14\omega - 123 = 0 200ω2+14ω−123=0It is given that the sequence of temperatures is a decreasing sequence. Find the value of T4T_4T4 and the value of T5T_5T5.
The limit of TnT_nTn as n→∞n \to \inftyn→∞ is LLL.
Write down an equation for LLL.
Find the value of LLL.
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.