Suites de nombres, cours, première ES - MathsFG

u n u(n)
u u(n) = 3nu(4) = 34= 81
•n u unu(n)
•unn
•un+1 unun−1un
•u(un)
•u0unn+ 1
•u1unn
f[0; +∞[R(un)
n∈Nun=f(n)n un
nf 7−→ un
(un)n un= 3n2+ 2
u1= 3 ×12+ 2 = 5
u2= 3 ×22+ 2 = 14
u10 = 3 ×102+ 2 = 302

fR R
u0u1
n∈N, un+1 =f(un)
u0
f
7−→ u1
f
7−→ u2
f
7−→ ... f 7−→ un
un=(u0= 4
un+1 = 2un+ 3
u1= 2u0+ 3 = 2 ×4 + 3 = 11
u2= 2u1+ 3 = 2 ×11 + 3 = 25
u3= 2u2+ 3 = 2 ×25 + 3 = 53
•(un)nn un+1 > un
•(un)nn un+1 < un
(un)
(n;un)

un= 4×1
2n
n
r r
u0u1n
un+1 =un+r
u0
+r
7−→ u1
+r
7−→ u2
+r
7−→ ... +r 7−→ un
u0= 11 un+1 =un−2n
u1=u0−2 = 11 −2=9
u2=u1−2=9−2=7
u3=u2−2=7−2=5
(un)n un+1 −un

(un)n∈Nun=−4+2n
n∈N, un+1 −un= (−4 + 2(n+ 1)) −(−4+2n) = −4+2n+ 2 + 4 −2n= 2
(un)
(un)
un=
−4+2n n
0 1 2 3 4 5 6 7−1−2 0
1
2
3
4
5
6
7
−1
−2
−3
−4
−5
(un)r
•r > 0 (un)
•r < 0 (un)