EMILE : Enseignement d'une Matière Intégrée à une Langue Étrangère In a factory, an oven bakes ceramics at a temperature of 1000° C. At the end of cooking, it is off and it cools. We are interested in the cooling phase of the oven, which starts from the moment it is turned off. The oven temperature is expressed in degrees Celsius (°C).The oven door can be safely opened for ceramics as soon as its temperature is below 70 ° C. Otherwise the ceramics can crack or even break.

For all integer n, we denote Tn the temperature in degrees Celsius after n hours from the moment the oven is turned off.

1. What is the value of T0?

The temperature is calculated by the following algorithm:

 Input n 1000←T For i=1 to n T←0.82×T+3.6 End For Display T
1. Work out the relation between Tn+1 and Tn.

2. Let (Vn) be the sequence defined by Vn = Tn - 20, for all n, .

1. Prove that  (Vn) is a geometric sequence. Give its first term and ratio.

2. Give Vn in terms of n.

3. Deduce Tn that for all n, .

3. After how many hours can the oven be opened safely for ceramics? You could find the solution of this question by writing an algorithm and/or by solving an inequation.

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