✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✕ DƯƠNG VĂN DŨNG P❍×❒◆● ❚❘➐◆❍ P❍■✲➷✲❚➷✲◆➷▼ ❈❻P ❍❆■ ❉❸◆● ❙❆■ P❍❹◆ ❱⑨ Ù◆● ❉Ö◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ✣➔ ◆➤♥❣ ✲ ✷✵✷✵ ✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✕ ❉×❒◆● ❱❿◆ ❉Ơ◆● P❍×❒◆● ❚❘➐◆❍ P❍■✲➷✲❚➷✲◆➷▼ ❈❻P ❍❆■ ❉❸◆● ❙❆■ P❍❹◆ ❱⑨ Ù◆● ❉Ö◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ P❤→♣ ❚♦→♥ ❙ì ❈➜♣ ▼➣ sè✿ ✽✳✹✻✳✵✶✳✶✸ ▲❯❾◆ ❱❿◆ ữớ ữợ ❈❤û ❱➠♥ ❚✐➺♣ ❚❙✳ ▲➯ ❍↔✐ ❚r✉♥❣ ✣➔ ◆➤♥❣ ✲ ✷✵✷✵ ▲❮■ ❈❷▼ ❒◆ ▲í✐ ✤➛✉ t✐➯♥ ❝õ❛ ❧✉➟♥ ✈➠♥ ❡♠ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ tỵ✐ ❤❛✐ t ữợ ỷ r t t ữợ tr♦♥❣ s✉èt q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ✤➸ ❡♠ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❊♠ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ♥❤➜t ✤➳♥ t➜t ❝↔ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✤➣ t➟♥ t➻♥❤ ❞↕② ❜↔♦ ❡♠ tr♦♥❣ sốt tớ t õ ỗ tớ ❡♠ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❝→❝ ❜↕♥ tr ợ Pữỡ P ỡ ◗✉↔♥❣ ❇➻♥❤✱ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈ø❛ q✉❛✳ ❚→❝ ❣✐↔ ❉×❒◆● ❱❿◆ ❉Ơ◆● ▼Ư❈ ▲Ư❈ ▼Ð ✣❺❯ ❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶ ✸ ✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷✳ ❙ü ❤ë✐ tư ❝õ❛ ♥❣❤✐➺♠ tỵ✐ ♥❣❤✐➺♠ t✉➛♥ ❤♦➔♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ❈❍×❒◆● ✷✳ ❚➑◆❍ ❈❍❻❚ ❍❐■ ❚Ư ❱⑨ ❙Ü ❚➬◆ ❚❸■ ◆●❍■➏▼ ❑❍➷◆● ❇➚ ❈❍➄◆✱ ◆●❍■➏▼ ❚❯❺◆ ❍❖⑨◆ ❈Õ❆ ▼❐❚ ▲❰P P❍×❒◆● ❚❘➐◆❍ P❍■✲➷✲❚➷✲◆➷▼ ❈❻P ❍❆■ ❉❸◆● ❙❆■ P❍❹◆✳ ✾ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✷✳✶✳ ❚r÷í♥❣ ❤đ♣ {An }∞ n=0 ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❧➔ ♠ët sè ❝❤➤♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷✳ ❚r÷í♥❣ ❤đ♣ {An }∞ n=0 ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❧➔ ♠ët sè ❧➫✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ❈❍×❒◆● ✸✳ ▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ Ù◆● ❉Ư◆●✳ ❑➌❚ ▲❯❾◆ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳✳✳✳✳✳✳✳✳✳✳ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✺ é ỵ t ỵ tt t ữỡ tr s ởt ữợ ự q trồ tr t➼❝❤ ✈➔ ù♥❣ ❞ö♥❣✳ ❈→❝ ✈➜♥ ✤➲ t✐➯✉ ❜✐➸✉ ♠➔ ỵ tt t ữỡ tr s q t ❧➔ t➼♥❤ ê♥ ✤à♥❤✱ t➼♥❤ t✉➛♥ ❤♦➔♥✱ t➼♥❤ ❞❛♦ ✤ë♥❣✱ t ợ t út ỵ tt ✤÷đ❝ ù♥❣ ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❝õ❛ t♦→♥ ❤å❝ ❝ơ♥❣ ♥❤÷ ❝→❝ ❦❤♦❛ ❤å❝ ❦❤→❝✱ ♥❤÷ tr♦♥❣ ❣✐↔✐ t số ỵ tt ỵ tt trỏ ỡ ỵ tt số t ỵ tt ỵ tt ữủ tỷ tr s t t t ỵ ự ỵ tt ❧➔ ♠ët ✈➜♥ ✤➲ t❤í✐ sü ❝õ❛ t♦→♥ ❤å❝ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ q✉❛♥ t➙♠✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙② ✤➣ ❝â ♥❤✐➲✉ t➔✐ ❧✐➺✉ ✈➔ ♥❤✐➲✉ ❜➔✐ ❜→♦ r ỵ t❤✉②➳t ✤à♥❤ t➼♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✱ ❝â ♥❤✐➲✉ ❜➔✐ t♦→♥ ❞➝♥ ✈➲ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✤à♥❤ t➼♥❤ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ♣❤✐✲æ✲tæ✲♥æ♠ ♣❤✐ t✉②➳♥ ❝â ❝❤➟♠✳ ❚➼♥❤ ❝❤➜t ❝õ❛ ♥❣❤✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ✤➣ ✤÷đ❝ ♥❤✐➲✉ t→❝ ❣✐↔ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♥â ❝ơ♥❣ ❧✉ỉ♥ ❧➔ ♠ët ✤➲ t➔✐ t❤✉ ❤ót sü q✉❛♥ t➙♠ ❝õ❛ ♥❤ú♥❣ ♥❣÷í✐ t ữợ sỹ ữợ ỷ ❚✐➺♣ ✈➔ ❚❙✳ ▲➯ ❍↔✐ ❚r✉♥❣✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ P❤÷ì♥❣ tr➻♥❤ ♣❤✐✲ỉ✲tỉ✲♥ỉ♠ ❝➜♣ ❤❛✐ ❞↕♥❣ s❛✐ ♣❤➙♥ ✈➔ ù♥❣ ❞ö♥❣✳ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ▼ö❝ t✐➯✉ ❝õ❛ ✤➲ t➔✐ ❧➔ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ t ởt ợ ữỡ tr s ổtổổ ❝➜♣ ❤❛✐ ❞↕♥❣ s❛✐ ♣❤➙♥ xn+1 = An xn−1 , n = 0, 1, 2, B n + xn ✭✶✮ tr♦♥❣ ✤â {An } ✈➔ {Bn } ❧➔ ❤❛✐ ❞➣② sè ❞÷ì♥❣ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð k + ✈➔ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x0 , xn ❧➔ ❝→❝ sè ❦❤æ♥❣ ➙♠✳ ✷ ✸✳ ✣è✐ tữủ ự t ởt ợ ữỡ tr➻♥❤ ♣❤✐ æ✲tæ✲♥æ♠ ❝➜♣ ❤❛✐ ❞↕♥❣ s❛✐ ♣❤➙♥✳ ✹✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ◆❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ♣❤✐ t✉②➳♥ ❝➜♣ ❤❛✐ ❝â ❞↕♥❣ ✭✶✮ ð tr➯♥✳ ✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❚❤✉ t❤➟♣ ❝→❝ t➔✐ ❧✐➺✉ s÷✉ t➛♠ ✤÷đ❝✱ ❝→❝ s→❝❤ ✈ð ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥✱ t➻♠ ❤✐➸✉ ❝❤ó♥❣ ✈➔ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ✈➲ ✤➲ t➔✐ t❤❡♦ ❤✐➸✉ ❜✐➳t ❝õ❛ ♠➻♥❤ ♥❣➢♥ ❣å♥✱ t❤❡♦ ❤➺ t❤è♥❣ ❦❤♦❛ ❤å❝✳ ✻✳ Þ ♥❣➽❛ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t✐➵♥ ❝õ❛ ✤➲ t➔✐✳ ✣➲ t➔✐ ❝â tr t ỵ tt ự õ t❤➸ sû ❞ö♥❣ ❧✉➟♥ ✈➠♥ ❧➔♠ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❞➔♥❤ ❝❤♦ s✐♥❤ ✈✐➯♥ ♥❣➔♥❤ ❚♦→♥ ✈➔ ♥❤ú♥❣ ♥❣÷í✐ ❦❤ỉ♥❣ ❝❤✉②➯♥ t♦→♥ ❝➛♥ ❝→❝ ❦➳t q✉↔ ❝õ❛ t♦→♥ ✤➸ ù♥❣ ❞ö♥❣ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ t❤ü❝ t✐➵♥ ❝õ❛ ♠➻♥❤✳ ✼✳ ❚ê♥❣ q✉❛♥ ❝➜✉ tró❝ ❧✉➟♥ ✈➠♥ ◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ữủ tr ỗ ữỡ r ❝á♥ ❝â ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥✱ t➔✐ ❧✐➺✉ t❤❛♠ ữỡ tự ỡ s ã ữỡ t tử sỹ tỗ t ổ t ởt ợ ữỡ tr ổtổổ s ã ữỡ ởt số t ự ã ì é t ợ ữỡ tr s t✉②➳♥ ♣❤✐✲ỉ✲tỉ✲♥ỉ♠ ❝➜♣ ✭❦✰✶✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣ xn+1 = F (xn , xn−1 , xn−1 , , xn−k ), n = 0, 1, 2, ✭✶✳✶✮ tr♦♥❣ ✤â F : (0, +∞) → [0, +∞) ❧➔ ❤➔♠ ❧✐➯♥ tö❝✳ ✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ✈➼ ❞ö ❈→❝ ✤à♥❤ ♥❣❤➽❛ ✈➔ ✈➼ ❞ö tr♦♥❣ ♣❤➛♥ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶✱ ✸✱ ✹✱ ✺❪✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ▼ët ❞➣② sè ❞÷ì♥❣ {xn}n≥−k ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✶✮ ♥➳✉ ♥â t❤ä❛ ♠➣♥ ✭✶✳✶✮ ✈ỵ✐ t➜t ❝↔ n ∈ N✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ◆❣❤✐➺♠ {xn}∞n=−k ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ t ợ ý p tỗ t ởt sè ♥❣✉②➯♥ p ≥ s❛♦ ❝❤♦ xn+p = xn , n ≥ −k ✭✶✳✷✮ ❚❛ ♥â✐ ♥❣❤✐➺♠ t✉➛♥ ❤♦➔♥ t❤❡♦ ❝❤✉ ❦➻ ❝ì sð p ♥➳✉ p ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ♥❤ä ♥❤➜t t❤ä❛ ♠➣♥ ✭✶✳✷✮ ❱➼ ❞ư ✶✳✶✳✸✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ xn+1 = , xn n = 0, 1, 2, , ✭✶✳✸✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ [x0 = α (α = 0) ] ❑❤✐ ✤â✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ ❝â ♥❣❤✐➺♠ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦➻ ✷ α, , α ✺✺ ❇➔✐ t♦→♥ ✸✳✾✳ ❈❤♦ ❞➣② sè {xn} ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ xn+1 cos 4n+1 π xn−1 = , n = 0, 1, nπ sin nπ cos + + x n 3 ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x0 ✈➔ x−1 ❧➔ ❝→❝ sè ❦❤æ♥❣ ➙♠✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ lim xn = n ự ợ t ữỡ tr tr ❝â ❞↕♥❣ ♥❤÷ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ {An } = cos nπ nπ 4n + + cos , ∀n π , {Bn } = sin 3 ❚❛ ❝â✿ {An} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ √ √ 3 A0 = , A1 = , A2 = 2 ✈➔ {Bn} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ B0 = 1, B1 = √ 3+1 , B2 = √ 3+1 ❱➻ max {A0, A1, A2} < ✈➔ {B0, B1, B2} ≥ ♥➯♥ t❤❡♦ ỵ t õ lim xn = n ✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇➔✐ t♦→♥ ✸✳✶✵✳ ❈❤♦ ❞➣② sè {xn } ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ xn+1 = cos (2n+1) π xn−1 + sin 2nπ + xn , n = 0, 1, ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x0 ✈➔ x−1 ❧➔ ❝→❝ sè ❦❤æ♥❣ ➙♠✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ lim xn = n→∞ ✺✻ ự ợ t ữỡ tr tr õ ♥❤÷ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ {An } = cos 2nπ (2n + 1) π , {Bn } = + sin , ∀n 3 ❚❛ ❝â✿ {An} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ 1 A0 = , A1 = 1, A2 = 2 ✈➔ {Bn} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ B0 = 2, B1 = √ √ 4− 3+4 , B2 = 2 ❱➻ max {A0, A1, A2} = ✈➔ {B0, B1, B2} t ỵ t õ lim xn = n→∞ ✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇➔✐ t♦→♥ ✸✳✶✶✳ ❈❤♦ ❞➣② sè {xn } ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝ xn+1 = + cos 4nπ xn−1 + sin (4n+1)π + xn , n = 0, 1, ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x0 ✈➔ x−1 ❧➔ ❝→❝ sè ❦❤æ♥❣ ➙♠✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ lim xn = n ự ợ t ữỡ tr➻♥❤ tr➯♥ ❝â ❞↕♥❣ ♥❤÷ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ {An } = + cos (4n + 1)π 4nπ , {Bn } = + sin , ∀n ❚❛ ❝â✿ {An} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ 1 A0 = 2, A1 = , A2 = 2 ✈➔ {Bn} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ 5 B = , B1 = , B2 = 2 ✺✼ ❱➻ A0A1A2 = 12 < ✈➔ {B0, B1, B2} = ỵ ✭✷✳✷✳✹✮ t❛ ❝â✿ lim xn = n→∞ ✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇➔✐ t♦→♥ ✸✳✶✷✳ ❈❤♦ ❞➣② sè {xn } ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ xn+1 2cos2 nπ xn−1 , n = 0, 1, = + cos 4nπ + 2x n ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x0 > ✈➔ x−1 = 0✳ ❈❤ù♥❣ ♠✐♥❤ r ữỡ tr õ t ợ ý ❝ì sð ❜➡♥❣ ✻✳ ❈❤ù♥❣ ♠✐♥❤✳ P❤÷ì♥❣ tr➻♥❤ tr➯♥ ✤÷đ❝ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉ xn+1 = cos2 nπ xn−1 = , n = 0, 1, cos2 2nπ + x n + xn cos2 nπ xn−1 1+cos 4nπ ◆❤➟♥ t❤➜② ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❝â ❞↕♥❣ ữ ữỡ tr ợ {An } = cos2 2n nπ , {Bn } = cos2 , ∀n 3 ❚❛ ❝â✿ {An} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ 1 A0 = 1, A1 = , A2 = 4 ✈➔ {Bn} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ 1 B0 = 1, B1 = , B2 = 4 ❱➻ BA BA AB = ♥➯♥ ❚❤❡♦ ✣à♥❤ ỵ t ữỡ tr õ t ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✻✳ ✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ 2 ✺✽ ❇➔✐ t♦→♥ ✸✳✶✸✳ ❈❤♦ ❞➣② sè {xn} ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ xn+1 = √ 2cos2 (2n+3)π xn−1 + sin (4n+1)π + xn , n = 0, 1, ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x0 ✈➔ x−1 ❧➔ ❝→❝ sè ❦❤æ♥❣ ➙♠✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♠å✐ ♥❣❤✐➺♠ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤ë✐ tư t↕✐ ♥❣❤✐➺♠ ❝â ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✻✳ ❈❤ù♥❣ ♠✐♥❤✳ ✈ỵ✐ ◆❤➟♥ t❤➜② ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❝â ❞↕♥❣ ♥❤÷ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ √ (4n + 1)π , ∀n {An } = 2cos2 , {Bn } = + sin ❚❛ ❝â✿ {An} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ √ A0 = 2, A1 = √ 2 , A2 = 2 √ ✈➔ {Bn} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ 5 B = , B1 = , B2 = 2 ❱➻ A0A1A2 = ✈➔ {B0, B1, B2} = ỵ t ữỡ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❤ë✐ tư t↕✐ ♥❣❤✐➺♠ ❝â ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✻✳ ✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇➔✐ t♦→♥ ✸✳✶✹✳ ❈❤♦ ❞➣② sè {xn } ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝ xn+1 + sin2 nπ xn−1 = , n = 0, 1, cos2 2nπ + x n ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x0 > ✈➔ x−1 = ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❦❤ỉ♥❣ ❤ë✐ tư✳ ❈❤ù♥❣ ♠✐♥❤✳ ✈ỵ✐ ◆❤➟♥ t❤➜② ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❝â ❞↕♥❣ ♥❤÷ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ {An } = + sin2 nπ 2nπ , {Bn } = cos2 , ∀n 3 ✺✾ ❚❛ ❝â✿ {An} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ 7 A0 = 1, A1 = , A2 = 4 ✈➔ {Bn} ❧➔ ❞➣② t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý ❝ì sð ❜➡♥❣ ✸ ✈ỵ✐ 1 B0 = 1, B1 = , B2 = 4 ❱➻ BA BA AB = 49 > ♥➯♥ ❚❤❡♦ ✣à♥❤ ỵ t t ữỡ tr õ ❦❤ỉ♥❣ ❤ë✐ tư✳ ✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ 2 ✻✵ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➲ ❝➟♣ ởt số t t ởt ợ ữỡ tr➻♥❤ ♣❤✐✲æ✲tæ✲♥æ♠ ❝➜♣ ❤❛✐ ❞↕♥❣ s❛✐ ♣❤➙♥✳ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ♠ët sè ❦➳t q✉↔ s❛✉ ❚r➻♥❤ ❜➔② ❤➺ t❤è♥❣ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ t t tử sỹ tỗ t ổ sỹ tỗ t t ợ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ♣❤✐ t✉②➳♥ ✤÷đ❝ ✤➲ ❝➟♣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ ❳➙② ❞ü♥❣ ✤÷đ❝ ♠ët sè ❜➔✐ t➟♣ ✈➲ ❞➣② sè ♠➔ ✈✐➺❝ ❣✐↔✐ q✉②➳t ❝❤ó♥❣ ❝â t❤➸ sû ❞ư♥❣ ❦➳t q✉↔ ❝õ❛ ❈❤÷ì♥❣ ✷✳ ✻✶ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ✣➦♥❣ ✣➻♥❤ ❈❤➙✉✱ ▲➯ ✣➻♥❤ ✣à♥❤✱ P❤❛♥ ❱➠♥ ❍↕♣✱ ▲➯ ✣➻♥❤ ❚❤à♥❤ ✭❈❤õ ❜✐➯♥✮ ✭✷✵✵✶✮✱ P❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ✈➔ ♠ët sè ù♥❣ ❞ö♥❣✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ö❝✳ ❬✷❪ ▲➯ ✣➻♥❤ ✣à♥❤ ✭✷✵✵✶✮✱ ❇➔✐ t➟♣ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ư❝✳ ❬✸❪ ▲➯ ✣➻♥❤ ✣à♥❤✱ ▲➯ ✣➻♥❤ ❚❤à♥❤ ✭✷✵✵✺✮✱ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ s❛✐ ♣❤➙♥✱ ◆❤➔ ①✉➜t ❜↔♥ ✣❍◗● ❍➔ ◆ë✐✳ ❚✐➳♥❣ ❆♥❤ ❬✹❪ ❊✳ ❈❛♠♦✉③✐s✱ ●✳ ▲❛❞❛s✱ ▲✳❲✳ ❘♦❞r✐❣✉❡s ❛♥❞ ❙✳ ◆♦rt❤s❢✐❡❧❞ ✭✶✾✾✹✮✱❖♥ βx t❤❡ r❛t✐♦♥❛❧ r❡❝✉rs✐✈❡ s❡q✉❡♥❝❡s xn+1 = 1+x ✱ ❈♦♠♣✉t❡rs ▼❛t❤ ❆♣♣❧✱ ✷✽✿ ✸✼✲✹✸✳ ❬✺❪ ❈✳ ●✐❜❜♦♥s✱ ●✳ ▲❛❞❛s✱ ▼✳❘✳❙✳ ❑✉❧❡♥♦✈✐❝ ✭✷✵✵✵✮✱ ❖♥ t❤❡ ❘❡❝✉rs✐✈❡ ❙❡✲ α+βx q✉❡♥❝❡ xn+1 = ✱ ▼❛t❤ ❙❝✐✳ ❘❡s✱ ✹✿ ✶✲✶✶✳ γ+x ❬✻❪ ●✳ ▲❛❞❛s ❛♥❞ ❚✳◆❡s❡♠❛♥♥✱ ❲✳❏✳ ❇r✐❞❡♥ ✭✶✾✾✾✮✱ ❖♥ t❤❡ ❇❡❝✉rs✐✈❡ ❙❡✲ A ✱ ❏✳ ❉✐❢❢❡r✳ ❊q✉❛t✐♦♥s ❆♣♣❧✱ ✺✿ ✹✾✶✲✹✾✹✳ q✉❡♥❝❡ xn+1 = max x ,x n n n−1 n n n n−1 DAr HQC DP NANG TRUONG DAI HOC , lhnP su PHAM So:v uIQD-DHSP CQNG HoA XA HOI CHD NGHiA VIET NAM DQc I,p - TI}'do - Hanh phuc t» Nang, t1t thang 1L nam J)lj QUYETDJNH vi vi~c giao di tai va trach nhiem huong dio lu,o van thac si HIEU TRUONG TRUONG DAI HOC SU PHAM Can ClI Nghi dinh s6 32/CP 04/411994 cua Chinh phu v~ viec l~p Dai h9C Da N~g; Can c,u Thong tu s6 08/20~4(TT-BGDDT 20/3/2014 cua BQ Giao due va Dao tao ve viec ban hanh Quy che to chirc va heat dong cua dai h9C vung va cac co so giao due dai h9Cthanh vien; Can cir Quyet dinh s6 6950IQD-DHDN 01112/2014 cua Giam d6c Dai hoc Da N~ng ban hanh Quy dinh nhiern vu, quyen han cua Dai hoc Da Nang, cac co so giao due dai hoc vien va cac don vi tnrc thuoc; Can cir Thong tu s6 15/2014/TT-BGDDT 15/5/2014 cua BQ Giao dl,lC va Dao t~o v~ vi~c ban hanh Quy ch~ Dao t~o trinh d9 th~c sl; Can Cll' Quy~t dinh 1060IQD-DHSP 0111112016 cua Hi~u tnroog Tnrong D~i hQc Su ph