Raisonnement par récurrence

Description

MATHS-LYCEE.FR RESSOURCES ET SOUTIEN MATHS AU LYCÉE
Application du raisonnement par récurrence

Informations

• Publié par: MATHS-LYCEE.FR
• Publié le: 22 septembre 2015
• Nombre de lectures: 15
• Langue: Français

Extrait

❈❤❛✐♣rt❡✶✿s✐✉❡ts
✕

rt❡♠✐♥❛❧❡✲❙♦❜❥❝❡❢✐t❇❈❆

▼❚❆❍❙✲▲Y❈❊❊✳❋❘

▼❆❚❍❙✲▲Y❈❊❊✳❋❘
❚❙✲r❡①❡❡❝✐❝r♦❝❈é❣✐✿♣✐❤❛❡✶tr
❈❤❛♣✐tr❡✶✿s♦♠♠❡❞❡st❡r♠❡s❞✬✉♥❡s✉✐t❡❡♥✉t✐❧✐s❛♥t❡❧r❛✐s♦♥♥❡♠❡♥t♣❛rré❝✉rr❡♥❝❡

❈❤❛♣r❡✐t✶✿

✳✶

✳✷

❖♥❝♦♥s✐❡r❞è❛❧✉st✐❡

❖♥♥♦t❡

❈❝❧❛✉r❡❧

☛❧✉❙♦♦♥t✐

❊♥♣❡r♥t❛♥

❊♥♣r❡♥t❛♥

▼rt♦♥❡r♣♦r✉t♦t✉❡✐❡♥tr♥❛t✉r❡❧

☛❧♦t✉❙♦✐♥

❊X❊❘❈■❈❊ ✶✼✲✷✲

❙✉✐t❡s

❡t♠♣sts❡✐♠✿é✶✵✲✶✷♠♥

2n−1
(un)❧❡r✉t❛♥r❞✜ért♦✉❡♣♥✐✐❡♥tt❡♦✉n♣r❛un=✳
n
2
n
Sn= Σuk=u0+u1+u2....+un
k=0
WWW✳▼❆❚❍❙✲▲Y❈❊❊✳❋❘
u0❡tu1✳

✿

u0=−1t❡

✿

n= 0✱ ♦♥ ❛

n= 1✱ ♦♥ ❛

1
u1=
2

−1
u0= =−1
0
2
2×1−1 1
u1= =
1
2 2

n✱ ♦♥ ❛

2n+ 3
Sn= 2−
n
2

2n+ 3
❖♥♥t❡♦Pn♣r❧❛♦étr✐♣Sn= 2−
n
2
■✲❛✐t✐♥❧✐s❛t✐♦♥
2W×0W+ 3W✳▼❆❚❍❙✲▲Y❈❊❊✳❋0❘
❖♥ ❛u0=−1❡tS0= 2−= 2−3 =−1✭r♣❡❛♣❧12 =✮
0
2
2n+ 3
❧❡ts❡✐①✉♥❡❡✐t♥r♥❛t❡✉r❧n❡❡❧q✉tSn= 2−n✮
✲❖♥✉s♣♣❡s♦q✐✬✉✭P❡✐❛r✈
n
2
2(n+ 1) + 3
t❡♦♥❡✈t✉♠❡rtr♦♥✮❡✐
✉q❡❧♦✬♥❛❧❛r♦sSn+1= 2−✭Pn+1❛r✈
n+1
2
2n+ 5
❝✬ts❡à✐❞r❡Sn+1= 2−
n+1
2

❙✐✉t❡s

P❛❣❡ ✶✴✷

▼❚❆❍❙✲▲Y❈❊❊✳❋❘t❡♠r✐♥❡❧❛❙

▼❆❚❍❙✲▲Y❈❊❊✳❋❘

t❡♠r✐♥❡❧❛✲❙♦❜❥❡✐❝❢t❇❆❈

✕
❈❤❛♣❡r✐t✶✿✉st✐s❡

❈❤❛♣❡rt✐✶✿✉s❡✐st
✕

t❡♠r✐♥❛❡❧✲❙♦❜❡❥t❝❢✐❇❈❆

▼❆❚❍❙✲▲Y❈❊❊✳❋❘

▼❚❆❍❙✲▲Y❈❊❊✳❋❘
❚❙❡✲❡①❝❡❝r❝✐r✐❣é♦

❈❤❛♣t✐❡r✿✶

❈❤❛♣✐tr❡✶✿

Sn+1=u0+u1+....+un+un+1

❙✐✉t❡s

=Sn+un+1
2n+ 3
= 2−+un+1
n
2
2n−1 2(n+ 1)−1 2n+ 2−1 2n+ 1
un=❞♦♥❝un+1== =
n n+1n+1n+1
2 22 2

❖♥ ❛ ❞♦♥❝ ✿
2n+ 32n+ 1
Sn+1= 2−+
n n+1
2 2
2(2n2+ 3)n+ 1
= 2−+
n n+1
2×2 2
4n2+ 6n+ 1
= 2−+
n+1n+1
2 2
2n+ 1−(4n+ 6)
= 2 +♥✐s❣❡−t♥❛❧❛❜❡❡❞❛✈r✐♦♥❛✈❡❝❞❡❢r❛❝t
n+1
2
2n+ 1−4n−6
= 2 +
n+1
2
−2n−5
= 2 +
n+1
2
WWW✳▼❆❚❍❙✲▲Y❈❊❊✳❋❘
2n+ 5
= 2−
n+1
2
❞♦❝♥❛❧♣r♦♣é✈r❛✐❡str✐t❣♥❛r✉❛❡n+ 1t✐♦sP❡❛✐✈r
n+1

❖♥❛❞♦♥❝♠r♦é♥t♣r❛ré✉❝r❡✉q❡♥❝❡

❙✐✉t❡s

2n+ 3
Sn= 2−♦✉rt♦✉t❡♥t✐❡r♥❛tr✉❧❡♣
n
2

WWW✳▼❆❚❍❙✲▲Y❈❊❊✳❋❘

P❛❣❡ ✷✴✷

▼❆❚❍❙▲✲Y❈❊❊❋✳❘♠r❡t♥✐❛❧❡❙

4n+ 6

n✳

▼❚❆❍❙✲▲Y❈❊❊✳❋❘

❡t♠r♥✐❧❛❡✲❙♦❜❡❥❢t✐❝❇❆❈

✕
❈❤❛✐♣❡rt✶✿✐✉ss❡t