BACCALAURÉAT GÉNÉRAL ET TECHNOLOGIQUE SESSION 2009 ÉPREUVE SPÉCIFIQUE MENTION « SECTION EUROPÉENNE OU DE LANGUE ORIENTALE »
Académies de Paris-Créteil-Versailles Binôme : Anglais / Mathématiques Sujet 1 SEQUENCES The first part of this page is a summary that can help you to do the exercise.
Arithmetic sequence A sequence {an } is an arithmetic sequence with common difference d if it can be written an = an −1 + d , where
n≥2. The nth term of an arithmetic sequence whose first term is a1 and common difference d can be written an = a1 + (n − 1)d . The sum Sn of the n first terms of the sequence with common difference d is
n a +a Sn = a1 + a2 + a3 + ... + ak + ... + an = n 1 n or S n = 2a1 + ( n − 1) d . 2 2 Geometric sequences A sequence {an } is a geometric sequence with common ratio r if it can be written an = an −1.r , where n ≥ 2 . The nth term of a geometric sequence whose first term is a1 and common ratio r can be written an = a1r n −1 .
The sum Sn of the n first terms of the sequence with common ratio r is
r n −1 Sn = a1 + a2 + a3 + ... + ak + ... + an = a1 × r −1 EXERCISE In a mountain resort, a cliff is going to be set up to create a climbing site. Two businesses A and B have been contacted and made the following offers: Business A: The first metre to be set up, costs 20 £, and each extra metre costs 5 £ more than the previous one. Business B: The first metre to be set up, costs 20 £, and each extra metre costs 10 % more than the previous one. Let un be the price of the nth metre to be set up and Sn be the price of the equipment of a cliff n metres high by business A. Let vn be the price of the nth metre to be set up and Rn be the price of the equipment of a cliff n metres high by business B. 1. a. Give u1 , u2 , u3 . b. Give a recursive rule for un . What is the nature of the sequence un ? Give an explicit rule for un . c. Give S1 , S 2 , S3 . Write Sn in terms of un . What can you say about Sn ? Write Sn in terms of n. 2. a. Give v1 , v2 , v3 . Round your result to the nearest pound. b. Give a recursive rule for vn . What is the nature of the sequence vn ? Give an explicit rule for vn . c. Give R1 , R2 , R3 . Write Rn in terms of vn . What can you say about Rn ? Write Rn in terms of n. 3. What is the price to set up a cliff 50 metres high by the two businesses? Round to the nearest pound. 4. Figure out using your graphing calculator when Business A is more interesting than Business B. Page : 1/1