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1.6.7 Generate a sequence by formula or recurrence (A-level only)

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Question 15

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​+84

where T1=1200T_1 = 1200T1​=1200 and ω\omegaω is a constant.

a.

Find an expression, in terms of ω\omegaω, for T2T_2T2​.

[1]
bi.

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=0
[3]
bii.

It is given that the sequence of temperatures is a decreasing sequence. Find the value of T4T_4T4​ and the value of T5T_5T5​.

[4]
ci.

The limit of TnT_nTn​ as n→∞n \to \inftyn→∞ is LLL.

Write down an equation for LLL.

[1]
cii.

Find the value of LLL.

[1]

1.6.7 Generate a sequence by formula or recurrence (A-level only) Questions

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