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Ann´e Scolaire 2012-2013 e CPGE Lyc´e R´f´rence Meknes e ee

Prof:ADIL MESKAF

ECT1

Fiche n◦ 1(Recurrences et sommes)

Exercice 1. Ecrire sans symbole
5
k=1

k2 ;

les expressions ci-dessous j=8 j 2 j=3 3j ;

n=5 n2 x2n+5
;
n=1 (−1) n 8 s2 s=2 s2 +1

.......................................................................................................................................................
Exercice 2. Ecrire les sommes suivantes avec le symbole
A = 25 + 35 + 45 + ............... + n5 ;B = 1 − a + a2 − a3 + ............. + (−1)n an
4
6
2n
2
C = a2 + a4 + a6 + ............. + a ;
2n
4 n D = 1 − 2 + 3 − 5 + ...........(−1)n+1 n+1 ;C = ln(1 × 2 × 3 × 4 × ...........n)
2
3
4
........................................................................................................................................................
Exercice 3. Calculer les sommes suivantes :
2n
2012 n+1 n
2
k=1 k(2k − 1)(k + 1);
945 3; Dn = k=0 (6k + 4k + 1); C = k=0 (2k + 1); Bn =
2
4
6
2n
2
3
En = 1 + 2 + 2 + 2 + ....... + 2 ; Fn = 1 − 3 + 3 − 3 + ....... + (−1)n 3n ; k+1 3
5×2j
Gn = n 10j ; Hn =
; In = (5 × 2n + 2 × 32n ; Jn = 2n 23k+1 × 3 4k j=0 k=3
3j+1

An =

...........................................................................................................................................................
Exercice 4.
1. V´rifier rapidement que les ´galit´s suivantes: e e e 2. Calculer alors:

1
1
=k k(k+1) n
1
k=1 k(k+1)



1 k+1 1
1
= 1(k k(k+1)(k+2) 2 n 1 k=1 k(k+1)(k+2 )

et

puis



2 k+1 +

1 k+2) ..........................................................................................................................................................
Exercice 5.
1. D´terminer trois r´els a,b,c tels que pour tout k ∈ N − 0; 1, e e

k−5 k(k2 −1

=

a k−1 +

b k +

c k+1 n k−5 k=2 k(k2 −1)

2. D´duire la valeur de la somme e ........................................................................................................................................................
Exercice 6. Soit (ak )k∈

0,n

une famille de nombres r´els. Justifier l’´galit´ suivante: e e e (

n k=1 ak ) − nan

Exercice 7. Montrer par recurrence que:
1. ∀n ∈ N∗ ,

n−1 k=0 (2k

+ 1) = n2

2. a ∈ R+ , ∀n ∈ N, (1 + a)n ≥ 1 + na
3.

n−1
1
k=0 k(k+1)

=

n n+1 4. ∀n ≥ 3, n! ≥ 2 × 3n−2
5. ∀n ∈ N∗ , (1 + 1 )(1 + 1 )(1 + 1 )...............(1 +
1
3
5

1
)
2n−1

>



2n + 1

6. Soit (un ) la suite d´finie par u0 = 2 et u1 = −2 et ∀n ∈ N, un+2 = −2un+1 + 3un ; e D´montrer que: ∀n ∈ N, un = 1 + (−3)n e 1

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