EMILE : Enseignement d'une Matière Intégrée à une Langue Étrangère

Every year, in September, Richard takes 8.5 tons of seaweed from the beaches of his commune. As of September 1st, 2018, there were 230 tons of seaweed on these beaches. Every year, between the 1st of October and the 1st of September following, the quantity of seaweed on these beaches increases of 4%. The quantity in tons of seaweed present on the beaches on September 1st of the year 2018+n, is represented by un. So, u0= 230.

1. Check by calculation that Richard will have 230.36 tons on the beaches on September 1st, 2019.

2. Explain why for all n , un+1 = 1.04un-8.84.

3. Let (vn) be the sequence defined by, for all n , vn = un - 221.

1. Prove that (vn) is a geometric sequence whose common ratio is 1.04. Give its first term.

2. Give vn in terms of n.

3. Deduce that for all n, un = 221 + 9×1.04n.

1. Will the amount of seaweed present on these beaches exceed 250 tons a day? If yes, specify after how many years this quantity will be reached.

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