Về phương trình tuyến tính với các số fibonacci

ĐẠI HỌC THÁI NGUYÊN TRƢỜNG ĐẠI HỌC KHOA HỌC  - ĐINH THỊ HUYỀN VỀ PHƢƠNG TRÌNH TUYẾN TÍNH VỚI CÁC SỐ FIBONACCI Chun ngành: Phƣơng pháp Tốn sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC PGS.TS NƠNG QUỐC CHINH THÁI NGUN - 2019 ✐ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ▼ð ✤➛✉ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶ ✷ ✹ ✷ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✈ỵ✐ ❝→❝ sè ❋✐❜♦♥❛❝❝✐ ✾ ✶✳✶ ✶✳✷ ✶✳✸ ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ✷✳✺ ✷✳✻ ✷✳✼ ✷✳✽ ❉➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ❞➣② ▲✉❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ✼✼✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ✽✵✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ tê♥❣ q✉→t✱ ❝→❝ ❦❤→✐ ♥✐➺♠ ✳ ✳ ❚r÷í♥❣ ❤đ♣ m = ✈➔ m = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚r÷í♥❣ ❤đ♣ tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚r÷í♥❣ ❤đ♣ x(i) < b, ✈ỵ✐ ♠å✐ i ✳ ✳ ✳ ✳ ✳ ✳ rữớ ủ tỗ t i x(i) ≥ b ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ët sè ❦➳t q✉↔ ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ S1 ✳ ✳ ✳ ✳ ✳ ❚r÷í♥❣ ❤đ♣ b ❧➫ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤ù♥❣ ỵ ỵ t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✻ ✼ ✾ ✶✸ ✶✻ ✷✷ ✷✺ ✷✾ ✸✸ ✸✻ ✸✽ ✸✾ ✶ ▲í✐ ❝↔♠ ì♥ rữợ t tổ ỷ t ỡ t P ổ ố ữợ tổ ❤♦➔♥ t❤➔♥❤ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❑❤✐ ❜➢t ✤➛✉ ♥❤➟♥ ✤➲ t➔✐ t❤ü❝ sü tæ✐ ❝↔♠ ♥❤➟♥ ✤➲ t➔✐ ♠❛♥❣ ♥❤✐➲✉ ♥ë✐ ❞✉♥❣ ♠ỵ✐ ♠➫✳ ❍ì♥ ♥ú❛ ✈ỵ✐ ✈è♥ ❦✐➳♥ t❤ù❝ ➼t ä✐ ♥➯♥ r➜t ❦❤â ✤➸ t✐➳♣ ❝➟♥ ✤➲ t➔✐✳ ▼➦❝ ❞ị r➜t ❜➟♥ rë♥ tr♦♥❣ ❝ỉ♥❣ ✈✐➺❝ ♥❤÷♥❣ ❚❤➛② ✈➝♥ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥ ✈➔ t➙♠ ❤✉②➳t tr♦♥❣ ữợ tổ tr sốt t❤í✐ ❣✐❛♥ tỉ✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ❚r♦♥❣ q✉→ tr➻♥❤ t✐➳♣ ❝➟♥ ✤➲ t➔✐ ✤➳♥ q✉→ tr➻♥❤ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ❚❤➛② ❧✉æ♥ t➟♥ t➻♥❤ ❝❤➾ ❜↔♦ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ❝❤♦ tæ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❈❤♦ ✤➳♥ ❜➙② ❣✐í ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ tỉ✐ ✤➣ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✱ ①✐♥ ❝↔♠ ì♥ ❚❤➛② ✤➣ ✤ỉ♥ ✤è❝ ♥❤➢❝ ♥❤ð tỉ✐✳ ❚ỉ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥ ✈➔ P❤á♥❣ ✣➔♦ t↕♦ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚ỉ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❝→❝ ❚❤➛②✱ ❈ỉ t t tr t ỳ tự qỵ ❝ơ♥❣ ♥❤÷ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ♥❤➜t ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚æ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr÷í♥❣ ❚❍P❚ ❍♦❛ ▲÷ ❆ ✲ ◆✐♥❤ ❇➻♥❤ ♥ì✐ tỉ✐ ❝ỉ♥❣ t→❝ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï tỉ✐ ❤♦➔♥ t❤➔♥❤ ❝ỉ♥❣ ✈✐➺❝ ❝❤✉②➯♥ ♠ỉ♥ t↕✐ ♥❤➔ tr÷í♥❣ ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❝❤÷ì♥❣ tr➻♥❤ ❤å❝ t➟♣ ❝❛♦ ❤å❝✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ♥❤ú♥❣ ♥❣÷í✐ ❦❤ỉ♥❣ ♥❣ø♥❣ ✤ë♥❣ ✈✐➯♥✱ ❤é trđ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ❝❤♦ tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ ✣✐♥❤ ❚❤à ❍✉②➲♥ ✷ ▼ð ✤➛✉ ▲❡♦♥❛r❞♦ P✐s❛♥♦ ❇♦❣♦❧❧♦ ✭❦❤♦↔♥❣ ✶✶✼✵ ✕ ỏ ữủ t ợ t r Ps t ữợ t ởt t ữớ ị ổ ữủ ởt sè ♥❣÷í✐ ①❡♠ ❧➔ ✧♥❤➔ t♦→♥ ❤å❝ t➔✐ ❜❛ ♥❤➜t t❤í✐ ❚r✉♥❣ ❈ê✧✳ ❋✐❜♦♥❛❝❝✐ ♥ê✐ t✐➳♥❣ tr♦♥❣ t❤➳ ❣✐ỵ✐ ❤✐➺♥ õ ổ tr ỵ số ❘➟♣ ð ❝❤➙✉ ❹✉✱ ✈➔ ✤➦❝ ❜✐➺t ❧➔ ❞➣② sè ❤✐➺♥ ✤↕✐ ♠❛♥❣ t➯♥ æ♥❣✱ ❞➣② ❋✐❜♦♥❛❝❝✐ tr♦♥❣ ❝✉è♥ s→❝❤ ▲✐❜❡r ❆❜❛❝✐✳ ❉➣② sè ❋✐❜♦♥❛❝❝✐ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ✈➫ ✤➭♣ ❝õ❛ ❦❤♦ t➔♥❣ ❚♦→♥ ❤å❝✳ ❉➣② ❋✐❜♦♥❛❝❝✐ ①✉➜t ❤✐➺♥ ✈➔ ❜✐➳♥ ❤â❛ ✈ỉ t➟♥ tr♦♥❣ tü ♥❤✐➯♥✱ ✈ỵ✐ r➜t ♥❤✐➲✉ t➼♥❤ ❝❤➜t ✤➭♣ ✈➔ ù♥❣ ❞ö♥❣ q✉❛♥ trå♥❣✳ ✣➳♥ ♥❛② ❝â r➜t ♥❤✐➲✉ ♠ð rë♥❣ ❝õ❛ ❞➣② ❋✐❜♦♥❛❝❝✐ ♥❤÷ ❞➣② k ✲❋✐❜♦♥❛❝❝✐✳✳✳ ❍➛✉ ❤➳t ♥❤ú♥❣ t➼♥❤ ❝❤➜t tèt ❝õ❛ ♥❤ú♥❣ ❞➣② ♥➔② ✤➲✉ ①✉➜t ♣❤→t tø ❞➣② ❋✐❜♦♥❛❝❝✐✳ ▼ët tỗ t s s ợ ▲✉❝❛s✳ ❉➣② ♥➔② ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ ✤➦❝ ❜✐➺t tr♦♥❣ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡✳ ❍❛✐ ❞➣② ♥➔② ❧➔ ❝❤ó♥❣ ❝â ♠è✐ ❧✐➯♥ ❤➺ ❝❤➦t ❝❤➩ ✈ỵ✐ ♥❤❛✉✳ ❚r♦♥❣ tü ♥❤✐➯♥ ❝â ♥❤✐➲✉ ❤✐➺♥ t÷đ♥❣✱ sü ✈➟t ①✉➜t ❤✐➺♥ trị♥❣ ✈ỵ✐ ❞➣② sè ❋✐❜♦♥❛❝❝✐✳ ❍➛✉ ❤➳t ❝→❝ ❜ỉ♥❣ ❤♦❛ ❝â sè ❝→♥❤ ❤♦❛ ❧➔ ♠ët tr♦♥❣ ❝→❝ sè ✸✱ ✺✱ ✽✳ ❙è ♥❤→♥❤ tø ♠ët ❝➙② ❦❤✐ ✤✐ tø ❣è❝ ❧➯♥ ♥❣å♥ ❝ơ♥❣ t❤÷í♥❣ t✉➙♥ t❤❡♦ ❞➣② ❋✐❜♦♥❛❝❝✐ tứ rỗ ✺✱ ✽✱ ✶✸ ♥❤→♥❤✳ ◆❤ú♥❣ ❝❤✐➳❝ ❧→ tr➯♥ ♠ët ♥❤➔♥❤ ụ tữỡ ự ợ số r ✈➠♥ ♥➔② ❝❤ó♥❣ t❛ ✤✐ t➻♠ ❤✐➸✉ ❝→❝ ❜➔✐ t♦→♥ r✐➯♥❣✱ ❜➔✐ t♦→♥ tê♥❣ q✉→t ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ tr♦♥❣ ✤â ❝→❝ ❤➺ ♥❣❤✐➺♠ ❧➔ ❝→❝ sè ❋✐❜♦♥❛❝❝✐✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ ❞➔♥❤ ✤➸ tr➻♥❤ ❜➔② ❧↕✐ sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ sè ❋✐❜♦♥❛❝❝✐ ✈➔ sè ▲✉❝❛s✱ ❣✐ỵ✐ t❤✐➺✉ ❤❛✐ ❜➔✐ t♦→♥ ✼✼✾ ✈➔ ✽✵✹ ✈➔ ❧í✐ ❣✐↔✐ ❝õ❛ ❤❛✐ ❜➔✐ t♦→♥ ♥➔②✳ ❈→❝ ❦➳t q✉↔ ✤➣ ❜✐➳t ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ✈✐➳t t❤❡♦ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✱ ❬✸❪✳ ❈❤÷ì♥❣ ✷ t❛ t➟♣ tr✉♥❣ ✤✐ t➻♠ ❤✐➸✉ ❜➔✐ t♦→♥ tê♥❣ q✉→t✱ ❧í✐ ❣✐↔✐ ❜➔✐ t♦→♥ ✸ tr♦♥❣ ❦❤✐ m = 3, tø ✤â ✤÷❛ r❛ ❞ü ✤♦→♥ ❧í✐ ❣✐↔✐ ❝❤♦ ❜➔✐ t♦→♥ tê♥❣ q✉→t✳ ❈ö t❤➸ tr♦♥❣ ♣❤➛♥ ✷✳✶ ❣✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ tê♥❣ q✉→t✳ P❤➛♥ ✷✳✷ tr➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ tr♦♥❣ tr÷í♥❣ ❤đ♣ m = ❤♦➦❝ 4✳ P❤➛♥ ✷✳✸ tr➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ ❝❤♦ tr÷í♥❣ ❤đ♣ tê♥❣ qt õ ỵ P ❤➳t ✷✳✽ ❧➔ ❝→❝ ❦➳t q✉↔ ①♦❛② q✉❛♥❤ ✈✐➺❝ ❝❤ù♥❣ ỵ t q ❜✐➳t ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ✈✐➳t t❤❡♦ t➔✐ ❧✐➺✉ ❬✹❪✳ ✹ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ❉➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ❞➣② ▲✉❝❛s ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❉➣② sè ỵ (Fn)n N ữủ ổ tự tr ỗ F0 = 0, F1 = 1, Fn+1 = Fn + Fn−1 , (n ≥ 1), ð ✤➙② Fn ❧➔ sè ❤↕♥❣ t❤ù n ❝õ❛ ❞➣② sè ❋✐❜♦♥❛❝❝✐✳ ❈→❝ sè ✤➛✉ t✐➯♥ ❝õ❛ ❞➣② ❋✐❜♦♥❛❝❝✐✿ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ứ tự tr ỗ t❛ ❝â Fn+2 − Fn+1 − Fn = 0, ✈ỵ✐ ♠å✐ n ≥ ❉♦ ✤â t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ x2 − x − = ❤❛② x2 = x + ữỡ tr ợ xn−1 t❛ ✤÷đ❝ xn+1 = xn + xn−1 ✭✶✳✶✮ ❘ã r➔♥❣ ♥➳✉ ϕ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ t❤➻ − ϕ ❝ô♥❣ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮✳ ❉♦ ✤â ϕn+1 = ϕn + ϕn−1 ✈➔ (1 − ϕ)n+1 = (1 − ϕ)n + (1 − ϕ)n−1 n ❱ỵ✐ ♠é✐ ❝➦♣ sè t❤ü❝ a, b✱ t❛ ✤➦t Fa,b (n) = aϕn + b(1 − ϕ) ✳ ❑❤✐ ✤â t➜t ❝↔ ❝→❝ ❤➔♠ ♥➔② t❤ä❛ tự tr ỗ ❈→❝ ❤➔♠ Fa,b (n) = aϕn + b(1 − ϕ)n ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ s✐♥❤✳ ❚r♦♥❣ ✣à♥❤ ♥❣❤➽❛ ❞➣② ❋✐❜♦♥❛❝❝✐✱ số ữủ ữợ tr ỗ sỷ ổ õ ❦❤➠♥✳ ▼➺♥❤ ✤➲ s❛✉ ✤➙② ❝❤♦ t❛ ❝ỉ♥❣ t❤ù❝ t÷í♥❣ ♠✐♥❤ ❝õ❛ ❞➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ❝ỉ♥❣ t❤ù❝ ❇✐♥❡t✳ ❈ỉ♥❣ t❤ù❝ ❇✐♥❡t ✤÷đ❝ sû ❞ư♥❣ ❤ú✉ ❤✐➺✉ tr♦♥❣ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ s❛✉ ♥➔②✳ ▼➺♥❤ ✤➲ ✶✳✶✳✸✳ ❉➣② sè ❋✐❜♦♥❛❝❝✐ ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ Fn = √ n 1+ − √ √ n 1− ❉➣② ▲✉❝❛s ❧➔ ♠ët ❞➣② sè ✤÷đ❝ ✤➦t t➯♥ ♥❤➡♠ ✈✐♥❤ ❞❛♥❤ ♥❤➔ t♦→♥ ❤å❝ ❋r❛♥☛❝♦✐s ➆❞♦✉❛r❞ ❆♥❛t♦❧❡ ▲✉❝❛s ✭✶✽✹✷✲✶✽✾✶✮✱ ♥❣÷í✐ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❞➣② sè ❋✐❜♦♥❛❝❝✐✱ ❞➣② sè ▲✉❝❛s ✈➔ ❝→❝ ❞➣② t÷ì♥❣ tü✳ ●✐è♥❣ ♥❤÷ ❞➣② ❋✐❜♦♥❛❝❝✐✱ ♠é✐ sè tr♦♥❣ ❞➣② ▲✉❝❛s ❜➡♥❣ tê♥❣ ❝õ❛ ❤❛✐ số trữợ õ số ỗ tữỡ ỳ sè ▲✉❝❛s ❧✐➲♥ ♥❤❛✉ s➩ ❤ë✐ tư ✤➳♥ ❣✐ỵ✐ ❤↕♥ ❜➡♥❣ t➾ ❧➺ ✈➔♥❣✳ ❚✉② ✈➟② ❦❤→❝ ✈ỵ✐ ❞➣② ❋✐❜♦♥❛❝❝✐✱ ❤❛✐ sè ✤➛✉ t✐➯♥ tr♦♥❣ ❞➣② ▲✉❝❛s ❧➔ L0 = ✈➔ L1 = ✭tr♦♥❣ ❞➣② ❋✐❜♦♥❛❝❝✐ ❧➔ ✵ ✈➔ ✶✮✳ ❈❤➼♥❤ ✈➻ t❤➳ ♠➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ sè ▲✉❝❛s s➩ ❦❤→❝ ✈ỵ✐ sè ❋✐❜♦♥❛❝❝✐✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ❈❤♦ r, s ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❦❤→❝ ❦❤æ♥❣✳ s ự ợ (r, s) ữủ ❧➔✿ u0 (r, s) = 0, u1 (r, s) = 1, un (r, s) = run−1 + sun−2 (n ≥ 2) ❚r♦♥❣ tr÷í♥❣ ❤đ♣ (r, s) = (1, 1) t❛ ❦➼ ❤✐➺✉ sè ❤↕♥❣ t❤ù n ❝õ❛ ❞➣② ❧➔ Ln ✈➔ ❣å✐ ♥❣➢♥ ❣å♥ ❧➔ ❞➣② ▲✉❝❛s✳ ❚÷ì♥❣ tü ♥❤÷ ❞➣② ❋✐❜♦♥❛❝❝✐✱ ❜➡♥❣ q✉② ♥↕♣ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❞➣② ▲✉❝❛s ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ s❛✉✳ ợ số ữỡ n, t ❝â Ln = √ 1+ n − √ 1− n ❚ø ▼➺♥❤ ✤➲ ✶✳✶✳✸ ✈➔ ▼➺♥❤ t õ ỵ s ỵ t❛ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝→❝ sè ❤↕♥❣ tê♥❣ q✉→t s ỵ ợ số ữỡ n > m, t õ FnLm = Fn+m + Fn−m ❱ỵ✐ ♠é✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ n t❛ ✤➦t F−n = (−1)n Fn ✈➔ Ln = (−1)n Ln ✶✳✷ ❇➔✐ t♦→♥ ✼✼✾ ◆➠♠ ✶✾✾✺✱ t↕♣ ❝❤➼ ✏❚❤❡ ❋✐❜♦♥❛❝❝✐ ◗✉❛rt❡r❧②✑ sè ✸✸✳✶ ✤➣ ❣✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ❇✳✼✼✾ ❝õ❛ ❆♥❞r❡✇ ❈✉s✉♠❛♥♦✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✤â ❧➔✿ ❚➻♠ ❝→❝ sè ♥❣✉②➯♥ a, b, c ✈➔ d t❤ä❛ ♠➣♥ < a < b < c < d s ỗ t tự s ú ợ số ữỡ n Fn = Fna + 6Fn−b + Fn−c + Fn−d ✭✶✳✷✮ ✣➣ ❝â ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ❦❤→❝ ♥❤❛✉ ❣û✐ ❧í✐ ❣✐↔✐ ✤➳♥ t↕♣ ❝❤➼ ✏❚❤❡ ❋✐❜♦♥❛❝❝✐ ◗✉❛rt❡r❧②✑✱ ❤➛✉ ❤➳t ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ❝❤➾ ❣û✐ ✤➳♥ ❧í✐ ❣✐↔✐ a = 2, b = 5, c = 6, d = ✈➔ ❦❤➥♥❣ ✤à♥❤ r➡♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ Fn = Fn−2 + 6Fn−5 + Fn−6 + Fn−8 ✭✶✳✸✮ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ q✉② ♥↕♣ t❤❡♦ n ❧➔ ✤ì♥ ❣✐↔♥✳ ❈❤➾ ❝â ❇r✉❝❦✲ ♠❛♥ ✈➔ ❋✐❣❣❤✐♦♥ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❝ö t❤➸ ✈➔ ❝❤➾ r❛ ❝→❝❤ t➻♠ a, b, c, d✳ ❚✉② ♥❤✐➯♥✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ ❝➟♥ ✈➔ ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ ❞÷í♥❣ ♥❤÷ ❦❤ỉ♥❣ ❝â t➼♥❤ ❦❤→✐ q✉→t✳ ❚❛ ❝â t❤➸ ❝â ❝❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ ✭✶✳✸✮ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② t n ữ s ợ n = t ữỡ tr tữỡ ữỡ ợ F8 = F6 + 6F3 + F2 + F0 ✣➥♥❣ t❤ù❝ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣ ✈➻ F8 = 21, F6 = 8, F3 = 2, F2 = 1, F0 = ●✐↔ sû ✤➥♥❣ t❤ù❝ ✤ó♥❣ ✈ỵ✐ ♠å✐ sè tü ♥❤✐➯♥ ≤ k ≤ n ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✭✶✳✸✮ ✤ó♥❣ ✈ỵ✐ k = n + ❚❤❡♦ ✣à♥❤ ♥❣❤➽❛ ❞➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ❣✐↔ t❤✐➳t q✉② ✼ ♥↕♣ t❛ ❝â Fn+1 = Fn + Fn−1 = (Fn−2 + 6Fn−5 + Fn−6 + Fn−8 ) + F(n−1)−2 + 6F(n−1)−5 + F(n−1)−6 + F(n−1)−8 = (Fn−2 + Fn−3 ) + (Fn−5 + Fn−6 ) + (Fn−6 + Fn−7 ) + (Fn−8 + Fn−9 ) = Fn−1 + 6Fn−4 + Fn−5 + Fn−7 ❱➻ ✈➟② t❛ ❝â Fn+1 = F(n+1)−2 + 6F(n+1)−5 + F(n+1)−6 + F(n+1)−8 ✶✳✸ ❇➔✐ t♦→♥ ✽✵✹ ◆ë✐ ❞✉♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✽✵✹ ❧➔✿ ❍➣② t➻♠ t➜t ❝↔ ❝→❝ sè ♥❣✉②➯♥ a, b, c ✈➔ d ✭✈ỵ✐ < a < b < c < d s ỗ t tự s ú ợ số ữỡ n Fn = Fn−a + 9342Fn−b + Fn−c + Fn−d ✭✶✳✹✮ ◆❣❛② s❛✉ ✤â✱ ♥➠♠ ✶✾✾✼✱ ▲✳❆✳●✳ ❉❡rs❡❧ ✤➣ ✤÷❛ r❛ ❧í✐ ❣✐↔✐ ❝õ❛ ❜➔✐ t♦→♥ ✽✵✹ tr♦♥❣ sè ✸✺✳✶ ✭✶✾✾✼✮ ❝õ❛ t↕♣ ❝❤➼ ❚❤❡ ❋✐❜♦♥❛❝❝✐ ◗✉❛rt❡r❧②✳ ▲í✐ ❣✐↔✐ ❝ư t❤➸ ♥❤÷ s❛✉✳ ❚ø ♥❤➟♥ ①➨t 9342 = 9349 − = L19 − L4 , ð ✤➙② Lk ❧➔ số s tự ỷ ỗ t tự ❣✐ú❛ ❝→❝ sè ❋✐❜♦♥❛❝❝✐ ✈➔ sè ▲✉❝❛s t❛ ❝â Fm+19 − Fm−19 = Fm L19 , Fm+4 + Fm−4 = Fm L4 ❚rø ✈➳ ✈ỵ✐ ✈➳ ❝õ❛ ✷ ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ♥❤➟♥ ✤÷đ❝ Fm+19 − Fm−19 − Fm+4 − Fm−4 = Fm (L19 − L4 ) ✣➦t n = m + 19✱ t❛ ♥❤➟♥ ✤÷đ❝ ✤➥♥❣ t❤ù❝ s❛✉ Fn = Fn−15 + 9342Fn−19 + Fn−23 + Fn−38 ◆❤÷ ✈➟② t❛ ❝â ❝→❝ sè tr➯♥ ❝➛♥ t➻♠ ❧➔✿ a = 15, b = 19, c = 23, d = 38✳ ❇è♥ sè tr➯♥ ❝❤➼♥❤ ❧➔ ♠ët ❧í✐ ❣✐↔✐ ❝õ❛ ❜➔✐ t♦→♥ ✽✵✹✳ ✽ ◆❤➟♥ ①➨t ✶✳✸✳✶✳ ✭✐✮ ❚r♦♥❣ t❤ü❝ t➳✱ ✈✐➺❝ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ✼✼✾ ✈➔ ✽✵✹✱ ❝❤➼♥❤ ❧➔ ✈✐➺❝ t➻♠ ❝→❝ sè ❋✐❜♦♥❛❝❝✐ t❤ä❛ ♠➣♥ ỗ tự õ ữỡ tr t t ợ ❧➔ ❝→❝ sè ❋✐❜♦♥❛❝❝✐✳ ✭✐✐✮ ❘ã r➔♥❣ t❛ ❝â t❤➸ t❤❛② ✤ê✐ ❤➺ sè ❝õ❛ ❤↕♥❣ tû t❤ù ✷ ❝õ❛ ỗ t tự tr t s ữủ ởt t ợ ợ ❣✐↔✐ ❦❤→❝ ♥❤❛✉✳ ❱➼ ❞ö ✶✳✸✳✷✳ ❩❡✐t❧✐♥ ✤➣ t➻♠ r❛ a = 2, b = 20, c = 40, d = ❧➔ ❧í✐ ❣✐↔✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ Fn = Fn−2 + 9349Fn−20 + Fn−40 + Fn−41 ❚r♦♥❣ ❝❤÷ì♥❣ s❛✉ ✭♥ë✐ ❞ư♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✮ ❝❤ó♥❣ t❛ s➩ ♥❣❤✐➯♥ ự ữỡ tr t t ợ ♥❣❤✐➺♠ ❧➔ ❝→❝ sè ❋✐❜♦♥❛❝❝✐✳ ✷✺ ❚❛ ❝â ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â ✈ỵ✐ ≤ i ≤ n − s❛♦ ❝❤♦ u(i) ≤ u (n − 1) < z tọ ỵ s ❧➔ ❦➳t q✉↔ ❝❤➼♥❤ ✤➛✉ t✐➯♥ ❝õ❛ t✐➳t ♥➔②✳ ✣à♥❤ ỵ số ữỡ m {b, x(1), x(3), , x(m)} ❧➔ ♠ët tố ợ ữỡ tr sû r➡♥❣ x(i) < b✱ ✈ỵ✐ ♠å✐ i✳ ❑❤✐ ✤â ố ợ ởt số ữỡ o t õ {x(1), x(3), x(4), , x(m)} = {b − o − 1, b − o, b − o + 2, , b − 1} ❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✹✳✺ ✭❛✮ ✈ỵ✐ n = m, z = b, u(i) = x(i) t❛ t❤➜② x(m) = z − 1✳ ◆➳✉ m = t❤➻ Fx(1) = Fb − F−x(3) = Fb − Fb−1 = Fb−2 s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ◆➳✉ m > ✈➔ trø ❝↔ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ❝❤♦ Fz−1 = Fb−1 t❛ ❝â Fb−2 = Fx(1) + F−x(3) + F−x(4) + + F−x(m−1) ❚❤❡♦ ❇ê ✤➲ ✷✳✹✳✺✱ x(i) < b − ✈ỵ✐ ♠å✐ i✳ ❉♦ ✤â ✈➻ m − ≥ →♣ ❞ö♥❣ ♠ët ❧➛♥ ♥ú❛ ❇ê ✤➲ ✷✳✹✳✺ ✭❛✮ ✈ỵ✐ n = m − 1, z = b − 2, u(i) = x(i) ❝❤♦ t❛ x (m − 1) = z − = b − ◆➳✉ m − ≥ ❝❤ó♥❣ t❛ ❝â t❤➸ trø ❝↔ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❝❤♦ Fb−3 ✈➔ →♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✹✳✺ t❛ ❝â x(i) < b − 4✳ ⑩♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✹✳✺ ✭❛✮ ✈ỵ✐ z = b − 4, u(i) = x(i), n = m − ❑❤✐ ✤â x (m − 2) = b − 5✳ ❚✐➳♣ tö❝ q✉→ tr➻♥❤ ♥➔②✱ ❜➡♥❣ q✉② ♥↕♣ t❛ ❝â Fb = Fx(1) + F−(b−o) + F−(b−o+2) + + F−(b−3) + F−(b−1) ❉♦ ✤â t❤❡♦ ❇ê ✤➲ ✷✳✸✳✼ t❤➻ x(1) = b − o − 1✳ ✷✳✺ rữớ ủ tỗ t i x(i) b r ♣❤➛♥ ♥➔② t❛ s➩ ❝❤➾ r❛ r➡♥❣ ♥➳✉ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✱ x(i) ≥ b ✈ỵ✐ i ♥➔♦ ✤â✱ t❤➻ t➟♣ {x(i) : x(i) > b} ❤♦➦❝ ❜➡♥❣ t➟♣ {b + 1} ❤♦➦❝ ❜➡♥❣ t➟♣ {b + 2, b + 4, , b + o + 1, b + o + 2}✳ ✷✻ ●✐↔ sû n, z, u(k), k = 1, 3, 4, , ❧➔ ❝→❝ sè ♥❣✉②➯♥ s❛♦ ❝❤♦ Fz = Fu(1) + F−u(3) + F−u(4) + + Fu(n) ởt ỗ t tự ợ tố ❝❤➤♥ ✈ỵ✐ u(1) < z, u(3) < u(4) < < u(n), n ≥ ✈➔ ✈ỵ✐ u(i) ≥ z ✈ỵ✐ i ≥ ♥➔♦ ✤â✳ ✣➸ t❤✉➟♥ t✐➺♥ ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② t❛ ❦➼ ❤✐➺✉✿ ▲❙❚✿ ❧➔ ✈➳ tr→✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❦❤✐ ❝❤✉②➸♥ t➜t ❝↔ ❝→❝ sè ❤↕♥❣ ➙♠ tø ♣❤➼❛ ❜➯♥ ♣❤↔✐ s❛♥❣ ❜➯♥ tr→✐✳ ❘❙❚✿ ❧➔ ✈➳ ♣❤↔✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❦❤✐ ❝❤✉②➸♥ t➜t ❝↔ ❝→❝ sè ❤↕♥❣ ➙♠ tø ♣❤➼❛ ❜➯♥ ♣❤↔✐ s❛♥❣ ❜➯♥ tr→✐✳ ❇ê ✤➲ ✷✳✺✳✶✳ ●✐↔ sû ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✻✮ ①↔② r❛✳ ❑❤✐ ✤â ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ ✤ó♥❣✳ ✭❛✮ u(j) > z ✈ỵ✐ j ♥➔♦ ✤â✳ ✭❜✮ ◆➳✉ u(j) − z ❧➔ sè ❧➫ t❤➻ ❣✐→ trà ❧ỵ♥ ♥❤➜t u(j) ≥ z ①↔② r❛ t↕✐ j = n ✭❝✮ ●✐↔ sû u(n) = z + o✳ ◆➳✉ o ≥ t❤➻ u (n − 1) = z + o − 1✳ ✭❞✮ ●✐↔ sû u(n) = z + o ✈➔ u (n − 1) = z + o − 1, ✈ỵ✐ o ≥ ❑❤✐ ✤â u (n − 2) < z + o − 2✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭❛✮ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ u(j) ≤ z ợ j tứ ữỡ tr t õ u(j) = z ✤è✐ ✈ỵ✐ ♠ët sè j ≥ ❉♦ ✤â tø ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✻✮ s✉② r❛ u(n) = z ✈➔ u(j) ≤ z − ✈ỵ✐ j < n✳ ❱➻ t❤➳ ❝❤➾ sè ❧➫ ❧ỵ♥ ♥❤➜t ❝õ❛ ✈➳ ❜➯♥ ♣❤↔✐ ❧➔ z − 1✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✸✳✻ t❛ ❝â Fz + Fz + Fi = Fu(1) + i∈K≤z−2 Fi < Fz−1 + Fz < Fz + Fz , i∈J≤z−1 ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥ ✈➻ tr♦♥❣ ✤â K ❧➔ t➟♣ ❝→❝ ❝❤➾ sè ❝❤➤♥✱ J ❧➔ t➟♣ ❝→❝ ❝❤➾ sè ❧➫ tr♦♥❣ ✭✷✳✻✮✳ ❉♦ ✤â u(j) > z ợ j õ ữỡ tr ✭✷✳✻✮ ✈➔ t❤❡♦ ✭❛✮ ❝❤➾ sè ❧ỵ♥ ♥❤➜t ①↔② r❛ ð ✈à tr➼ n✳ ●✐↔ sû u(n) = z + o + 1, ♥❣❤➽❛ ❧➔ u(n) − z ❧➔ sè ❝❤➤♥✳ ❑❤✐ ✤â✱ ❝❤➾ sè ❧➫ ❧ỵ♥ ♥❤➜t ð ♣❤➼❛ ❜➯♥ ♣❤↔✐ ❧➔ z + o✳ ❉♦ ✤â s❛✉ ❦❤✐ ❝❤✉②➸♥ ❝→❝ sè ➙♠ s❛♥❣ ✈➳ ❜➯♥ tr→✐ ❝❤ó♥❣ t❛ ❝â RST = Fu(1) + S (z + o) < Fz + Fz+o+1 ≤ LST ♠➙✉ t❤✉➝♥✳ ❱➻ ✈➟② u(n) − z ♣❤↔✐ ❧➔ sè ❧➫✳ ✭❝✮ ❈❤♦ u(n) = z + o ✈➔ ❣✐↔ sû ♥❣÷đ❝ ❧↕✐ u (n − 1) = z + o − 1✳ ❚ø ♣❤÷ì♥❣ ✷✼ tr➻♥❤ ✭✷✳✻✮ t❛ ❝â u(j) < z + o − ✈ỵ✐ ♠å✐ j ≤ n − 1✳ ❱➻ z ❝❤➤♥✱ ❝❤➾ sè ❝❤➤♥ ❧ỵ♥ ♥❤➜t ❝õ❛ ✈➳ ❜➯♥ ♣❤↔✐ ❧➔ z + o − 3✳ ❉♦ ✤â✱ s❛✉ ❦❤✐ ❝❤✉②➸♥ ❝→❝ sè ➙♠ s❛♥❣ ❜➯♥ tr→✐ ✈➔ sû ❞ö♥❣ ❣✐↔ t❤✐➳t o ≥ t❛ ❝â LST = Fz + S (z + o − 3) < Fz + Fz+o+2 ≤ Fz+o ≤ RST, ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â u (n − 1) = z + o − 1✳ ✭❞✮ ❈❤♦ u(n) = z + o✱ u (n − 1) = z + o − ✈➔ ❣✐↔ sû ♥❣÷đ❝ ❧↕✐ u (n − 2) = z + o − 2✳ ❚ø ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✻✮✱ ❝❤➾ sè ❝❤➤♥ ❧ỵ♥ ♥❤➜t ❝õ❛ ✈➳ ❜➯♥ ♣❤↔✐ ❝❤♦ j ≤ n − ❧➔ z + o − 3✳ ❱➻ u(i) ≥ z ♥➯♥ ✤è✐ ✈ỵ✐ i ≥ ♥➔♦ ✤â F−u(n) + F−u(n−1) + F−u(n−2) = 2Fz+o−2 ❉♦ ✤â s❛✉ ❦❤✐ ❝❤✉②➸♥ ❝→❝ sè ➙♠ s❛♥❣ ❜➯♥ ♣❤↔✐ t❛ ❝â LST = Fz + S (z + o − 3) < Fz + Fz+o−2 < 2Fz+o−2 ≤ RST, ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â u (n 2) < z + o ỵ ✷✳✺✳✷✳ ❈❤♦ b, x(1), x(3), x(4), , x(m), m ❧➔ ♠ët ♥❣❤✐➺♠ ♥❣✉②➯♥ tè✱ ❧ỵ♥ ✈➔ ❝❤➤♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✳ ●✐↔ sû r➡♥❣ x(i) ≥ b ✈ỵ✐ ♠ët sè i = ♥➔♦ ✤â✳ ❑❤✐ ✤â ❤♦➦❝ ≥ x(m) = b + ✭✷✳✼✮ ❤♦➦❝ x(m) = b + o + 2, x (m − 1) = b + o + 1, x (m − 2) = b + o − 1, , x (m − j) = b + 2, ✭✷✳✽✮ ợ j ởt số ữỡ õ ự ♠✐♥❤✳ ⑩♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✺✳✶ ✭❜✮✱ ✈ỵ✐ z = b, u(j) = x(j), n = m, t❛ ❝â x(m) = b + o✳ ◆➳✉ o = t❤➻ ✭✷✳✼✮ ❧➔ ✤ó♥❣✳ ◆➳✉ o > t❤➻ ✈➻ o ❧➔ sè ❧➫ ♥➯♥ o ≥ 3✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✺✳✶ ✭❝✮✱ ✈ỵ✐ z = b, u(j) = x(j), n = m, t❛ ❝â x (m − 1) = b + o − ❚÷ì♥❣ tü →♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✺✳✶ ✭❞✮ t❛ ❝â x (m − 2) < b + o − 2✳ ❱➻ b ❧➔ sè ❝❤➤♥ ✈➔ o ❧➔ sè ❧➫ ♥➯♥ t❛ ❝â F−x(m) + F−x(m−1) = F−(b+o) + F−(b+o−1) = F−(b+o−2) ✷✽ ❉♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ✤÷đ❝ rót ❣å♥ t❤➔♥❤ Fb = Fx(1) + F−x(3) + F−x(4) + + F−x(m−2) + F−(b+o−2) ◆➳✉ o = t❤➻ x(m) = b + o = b + 3, x (m − 1) = b + o − = b + 2✳ ❉♦ ✤â ✭✷✳✽✮ ❧➔ t❤ä❛ ♠➣♥ ✈ỵ✐ o = ✈➔ j = 1✳ ◆➳✉ o ≥ t❤➻ ✈➻ o < x(1) < b ✈➔ x(3) < x(4) < < x (m − 2) < b + o − ♥➯♥ →♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✺✳✶✱ ✈ỵ✐ z = b, u(i) = x(i), i = 1, 3, 4, , m − 2, u (m − 1) = b + o − 2, n = m − 1, t❛ ✤÷đ❝ x (m − 2) = b + o − ✈➔ x (m − 3) < b + o − 4✳ ❚✐➳♣ tö❝ q✉→ tr➻♥❤ ✈➔ →♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✺✳✶ ❝❤♦ ✤➳♥ ❦❤✐ x (m − j) = b + ✈ỵ✐ ♠ët sè j ♥➔♦ ✤â✱ t❛ ❝â x(m) = b+o, x (m − 1) = b+o−1, x (m − 2) = b+o−3, , x (m − j) = b+2 ✣➦t o = o − t❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❙û ❞ö♥❣ ❇ê ✤➲ ✷✳✺✳✶ t❛ ❝â ♠ët ❤➺ q✉↔ q✉❛♥ trå♥❣ s❛✉✳ ❍➺ q✉↔ ✷✳✺✳✸✳ ●✐↔ sû tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ x(i) ≥ b ✈ỵ✐ ♠ët sè i ♥➔♦ ✤â✳ ❑❤✐ ✤â ✈ỵ✐ sè ♥❣✉②➯♥ j0 ♥➔♦ ✤â✱ t❛ ❝â ✭✐✮ k∈J F−x(k) = Fb+1 ✈ỵ✐ J = {j0 + 1, j0 + 2, , m}✳ ✭✐✐✮ ◆➳✉ k ❦❤æ♥❣ t❤✉ë❝ t➙♣ J t❤➻ x(k) ≤ b✳ ✭✐✐✐✮ x (j0) = b✳ ✭✐✈✮ {x(1), x(3), , x(m)} = S1 ∪ {x(j0)} ∪ S2 ✈ỵ✐ S1 = {x(j) : j < j0} ✈➔ S2 = {x(j) : j > j0} ✈ỵ✐ Fx(1) + j∈S −{x(1)} F−x(j) = Fb−2✱ x (j0) = b ✈➔ j∈S2 F−x(j) = Fb+1 ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✸✳✼ ❝❤♦ ✭✷✳✽✮ t❛ ❝â ✭✐✮✳ ✭✐✐✮ ❚❤❡♦ ❣✐↔ t❤✐➳t ❝õ❛ ỵ t r ◆➳✉ ✭✷✳✼✮ ❧➔ ✤ó♥❣✱ t❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ t❛ ❝â x(k) ≤ b✱ ✈ỵ✐ k ≤ m − ♥➔♦ ✤â✳ ◆➳✉ ✭✷✳✼✮ ❧➔ ✤ó♥❣ t❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✱ x(k) ≤ b + ✈ỵ✐ k ≤ j0 ✳ ❉♦ ✤â ✤➸ ❝❤ù♥❣ ♠✐♥❤ x(k) < x (j0 ) ≤ b ợ k j0 t sỷ ữủ r➡♥❣ x (j0 ) = b + ❚❤❡♦ ✭✐✮ t❛ ❝â t❤➸ ✈✐➳t ❧↕✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ♥❤÷ s❛✉ Fb = Fx(1) + F−x(3) + F−x(4) + + F−x(j0 +1) + F−x(j0 ) + Fb+1 ✭✷✳✾✮ ✷✾ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✸✳✻ ❝❤♦ ✭✷✳✾✮✱ t❛ ❝â Fb + Fb+1 > Fb + k∈K≤b Fi = k∈L Fk + Fb+1 +Fb+1 > 2Fb+1 , ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â x(k) ≤ b ✭✐✐✐✮ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ x(k) = b ✈ỵ✐ ♠å✐ k ✳ ✣➦t J ✈➔ j ♥❤÷ tr♦♥❣ ✭✐✮✳ ⑩♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✸✳✻ ❝❤♦ ✭✷✳✾✮ t❛ ✤÷đ❝ Fb+1 = Fb + Fb−1 > Fb + k∈K≤b−2 Fi = Fx(1) + k∈K Fi + Fb+1 > Fb+1 , ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â tỗ t j0 s x (j0 ) = b ✭✐✈✮ ✣➠t j0 ♥❤÷ tr♦♥❣ ✭✐✐✐✮✳ ✣à♥❤ ♥❣❤➽❛ ❝→❝ t➟♣ ❝♦♥ S1 ✈➔ S2 t❤❡♦ ❝→❝ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱ j < j0 , j = ✈➔ j > j0 t÷ì♥❣ ù♥❣✳ ❚❤❡♦ ✭✐✮✱ j∈S2 F−x(j) = Fb+1 ✈➔ t❤❡♦ ✭✐✐✐✮ F−x(jo ) = F−b ✳ ❉♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ✤÷đ❝ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉✿ Fb = Fx(1) + j∈S1 F−x(j) = Fb−2 , j∈S2 F−x(j) = Fb+1 ✷✳✻ ▼ët sè ❦➳t q✉↔ ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ S1 ▼ư❝ ✤➼❝❤ ❝õ❛ ♣❤➛♥ ♥➔② ❧➔ ♠ỉ t↔ ✤➛② ✤õ ❝➜✉ tró❝ ❝õ❛ t➟♣ S1 ✤÷đ❝ ①→❝ ✤à♥❤ tr♦♥❣ q r q t õ ỗ ♥❤➜t t❤ù❝ Fx(1) + F−x(j) = Fb−2 ✭✷✳✶✵✮ j∈S1 −{x(1)} ✣➦t k = inf {i : S1 ≤ x(i)}✳ ❚❤❡♦ ❍➺ q✉↔ ✷✳✺✳✸ ✭✐✈✮✱ t❛ ❝â ♥➳✉ k = t❤➻ x(1) = b − 2✳ ❉♦ ✤â tr♦♥❣ ♣❤➛♥ ❝á♥ ❧↕✐ ❝❤ó♥❣ t❛ ❣✐↔ sû k ≥ 3✳ ❑❤✐ ✤â k = j − 1✳ ❇ê ✤➲ ✷✳✻✳✶✳ ◆➳✉ x(i) ≤ b − 3✱ ✈ỵ✐ ≤ i ≤ k, t❤➻✿ {x(1), x(3), x(4), , x(k)} = {b − o − 3, b − o − 2, b − o, , b 3} ự ỵ ✷✳✹✳✻ ❜➡♥❣ ❝→❝❤ t❤❛② b ❜ð✐ b − ✈➔ m ✤÷đ❝ t❤❛② ❜ð✐ k ❑❤✐ ✤â ❝→❝ ❣✐↔ t❤✐➳t ❝õ❛ ❜ê ✤➲ ✷✳✻✳✶ ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➻✿ ✸✵ ✭✐✮ tt ỵ ợ m ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➻ k ≥ 3✳ ✭✐✐✮ b − ❧➔ sè ❝❤➤♥ ✈➻ b ❧➔ sè ❝❤➤♥✳ ✭✐✐✐✮ x(1) ≤ b − = (b − 2) − t❤❡♦ ❣✐↔ t❤✐➳t✳ ✭✐✈✮ ❇➜t ✤➥♥❣ t❤ù❝ ≤ x(i) ≤ b − s✉② r❛ ≤ b b > ỗ t tự ổ r t ỗ t tự ❝ơ♥❣ ❦❤ỉ♥❣ ①↔② r❛✳ ❉♦ ✤â t❤❡♦ ✣à♥❤ ❧➼ ✷✳✹✳✻ t❤❛② b ❜ð✐ b − t❛ ❝â ✭✷✳✶✶✮✳ ❇ê ✤➲ ✷✳✻✳✷✳ ●✐↔ sû ✈ỵ✐ ♠ët sè i, ≤ i ≤ k ♥➔♦ ✤â ✭✷✳✶✷✮ ❚❛ ❝â ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ ✤ó♥❣✳ ✭✐✮ x(i) = b − ✈ỵ✐ ♠ët sè i ♥➔♦ ✤â✳ ✭✐✐✮ ❑❤ỉ♥❣ t❤➸ ❝â x(1) = b − = x(k)✳ ✭✐✐✐✮ ❑❤æ♥❣ t❤➸ ❝â x(1) = b − ✈➔ x(j) ≤ b − 2✱ ✈ỵ✐ ≤ j ≤ k✳ ✭✐✈✮ x(k) = b − 1✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ x(i) ≤ b − ✈ỵ✐ ♠å✐ i✳ ❚❤❡♦ ✤✐➲✉ ❦✐➺♥ x(i) ≥ b − ✭✷✳✶✷✮ t❛ ❝â ♥❣❛② x(i) = b − ✈ỵ✐ ♠ët sè i ♥➔♦ ✤â✳ ❈❤ó♥❣ t❛ ❦❤ỉ♥❣ t❤➸ ❝â i = ữỡ tr Fx(1) = Fb2 t ủ ợ k ≥ ♠➙✉ t❤✉➝♥ ✈ỵ✐ x(1) ≤ b − = (b − 2) − tr♦♥❣ ❇ê ✤➲ ✷✳✻✳✶✳ ❱➟② x(i) = b − ✈ỵ✐ i ≥ ♥➔♦ ✤â✳ ❉♦ ✤â tø ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✱ t❛ ❝â i = k ✈➔ x(j) ≤ b − ✈ỵ✐ j < k ✳ ❙û ❞ö♥❣ ❇ê ✤➲ ✷✳✸✳✻ ❝❤♦ ✭✷✳✶✵✮ t❛ ❝â Fb−2 + Fb−2 + i∈J Fi = Fx(1) + i∈K≤b−3 Fi < Fb−3 +Fb−2 ≤Fb−2 + Fb−2 , t tỗ t i ♥➔♦ ✤â ✤➸ x(i) = b − 1✳ ✭✐✐✮ ●✐↔ sû ♥❣÷đ❝ ❧↕✐✱ t❤➻ Fb−2 = Fb−1 + F−x(3) + + F−x(k−1) + Fb−1 ✳ ❙û ❞ư♥❣ ❇ê ✤➲ ✷✳✸✳✻ t❛ ✤÷đ❝ Fb−1 + Fb−1 + i∈J Fi = Fb−2 + i∈K≤b−2 Fi < Fb−2 +Fb−1 , ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥✳ ❱➻ ✈➟② ❦❤æ♥❣ t❤➸ ①↔② r❛ x(1) = b − = x(k)✳ ✭✐✐✐✮ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ x(1) = b − ✈➔ x(i) = b − ✈ỵ✐ j ≥ 3✳ ❚rø Fb−2 tø ❝↔ ❤❛✐ ♣❤➼❛ ❝õ❛ ✭✷✳✶✵✮ t❛ ❝â = Fb−3 + F−x(3) + F−x(4) + + F−x(k) , ✸✶ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❍➺ q✉↔ ✷✳✹✳✹✳ ✭✐✈✮ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ♥➳✉ i ∈ S1 t❤➻ x(i) < b✳ ❚❤❡♦ ữỡ tr x(i) ợ t r i = k ❤♦➦❝ i = 1✳ ❉♦ ✤â tø ✭✐✮✱ ✭✐✐✮ ✈➔ ✭✐✐✐✮ t❛ ❝â x(k) = b − 1✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✻✳✷ ✭✐✈✮ ❝â ♠ët sè ♥❣✉②➯♥ ❧ỵ♥ ♥❤➜t p✱ ✈ỵ✐ ≤ p ≤ k − s❛♦ ❝❤♦ {x(k), x (k − 1) , , x (k − (p − 1))} = {b − 1, b − 2, , b − p} ✭✷✳✶✸✮ ❚❛ t❤➜②✱ ♥➳✉ k − (p − 1) > 3, t❤➻ tø t➼♥❤ ❝❤➜t ❧ỵ♥ ♥❤➜t ❝õ❛ p✱ t❛ ❝â x (k − p) ≤ b − p − ❚÷ì♥❣ tü✱ ♥➳✉ k − (p − 1) = t❤➻ x(1) ≤ b − p − ❚❤➟t ✈➟②✱ ❝❤♦ k − (p − 1) = ✈➔ ❣✐↔ sû ♥❣÷đ❝ ❧↕✐ r➡♥❣ x(1) = b − p − 2✳ ❚❤❛② ✭✷✳✶✸✮ ✈➔♦ ✭✷✳✶✵✮ t❛ ✤÷đ❝ Fb−2 = Fb−1 − Fb−2 + Fb−3 − Fb−4 , ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❇ê ✤➲ ✷✳✸✳✹✳ ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝➜✉ tró❝ ❝õ❛ S1 ❜➡♥❣ ❝→❝❤ ①➨t ❜❛ trữớ ủ p tữợ k (p − 1)✳ ❇ê ✤➲ ✷✳✻✳✸✳ ●✐↔ sû ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✸✮ ❧➔ ❝è ✤à♥❤✳ ❑❤✐ ✤â ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ ✤ó♥❣ ✭✐✮ ◆➳✉ p ❧➔ ❝❤➤♥ ✈➔ k − (p − 1) = t❤➻ {x(1), x(3), x(4), , x(k)} = {b − o − 3, b − o − 1, b − o, , b − 1} ✭✷✳✶✹✮ ✭✐✐✮ ◆➳✉ p ❧➔ ❝❤➤♥ ✈➔ k − (p 1) > t ợ ởt số ữỡ j ♥➔♦ ✤â✱ {x(1), x(3), x(4), , x(j), x (j + 1) , x (j + 2) , , x(k)} = {b − o − − o , b − o − − o , b − o − − o , , b − o − 4} ✭✷✳✶✺✮ ∪ {b − o − 1, b − o, , b − 1} ✭✐✐✐✮ p ❦❤æ♥❣ t❤➸ ❧➔ sè ❧➫✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â Fx(1) = Fb−2 − F−x(k) + F−x(k−1) + + F−x(k−(p−1)) , ✭ ❜ð✐ ✭✷✳✶✵✮✮ = Fb−2 − (Fb−1 − Fb−2 ) + (Fb−3 − Fb−4 ) + + Fb−(p−1) − Fb−p = Fb−2 − Fb−3 − Fb−5 − Fb−7 − − Fb−(p+1) = Fb−(p+2) ✭t❤❡♦ ❇ê ✤➲ ✷✳✸✳✹✮, ✸✷ ð ✤➙② ❞➜✉ ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐ ❧➔ ❞♦ ✭✷✳✶✸✮✳ ✣➦t o = p + t❤➻ t❛ ❝â ✭✷✳✶✹✮✳ ✭✐✐✮ ❚÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ♣❤➛♥ ✭✐✮ t❛ ❝â Fx(1) + F−x(3) + F−x(4) + + F−x(k−p) = Fb−2 − F−x(k) + F−x(k−1) + + F−x(k−(p−1)) = Fb−2 − (Fb−1 − Fb−2 ) + (Fb−3 − Fb−4 ) + + Fb−(p−1) − Fb−p = Fb−2 − Fb−3 − Fb−5 − Fb−7 − Fb(p+1) = Fb(p+2) rữợ t t r❛ r➡♥❣ x(j) ≤ b − (p + 3) ✈ỵ✐ ≤ j ≤ k − p✳ ❉♦ t➼♥❤ tè✐ ✤↕✐ ❝õ❛ p tr♦♥❣ ✭✷✳✶✸✮ t❛ ❝â x(j) ≤ b − (p + 2) ✈ỵ✐ ≤ j ≤ k − p✳ ❍ì♥ ♥ú❛✱ ♥➳✉ x (k − p) = b (p + 2) ú t ỗ t tự s ổ ú ỗ t t❤ù❝ ✭✷✳✶✵✮ s➩ ❦❤æ♥❣ ①↔② r❛✳ ❱➻ t❤➳ x(j) ≤ b − (p + 3)✳ ❚✐➳♣ t❤❡♦ t❛ ❝❤➾ r❛ r➡♥❣ x(1) ≤ b − (p + 3)✳ ◆➳✉ x(1) = b − (p + 2) t❤➻ ✭✷✳✶✻✮ s➩ ❦❤æ♥❣ ✤ó♥❣✳ ▼➦t ❦❤→❝ ♥➳✉ x(1) ≥ b − (p + 1) t❤➻ ❇ê ✤➲ ✷✳✸✳✻ ❝❤♦ ✭✷✳✶✻✮ t❛ ❝â Fx(1) + i∈J Fi = i∈K≤b−(p+2) Fi < Fb−(p+1) ≤Fx(1) , ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❱➟② x(1) ≤ b − (p + 3)✳ ❱➻ x(1) ≤ b − (p + 3) ợ i ỵ ❝❤♦ ✭✷✳✶✻✮ ✈ỵ✐ b − (p + 2) t❤❛② t❤➳ ❝❤♦ b✱ s✉② r❛ ✈ỵ✐ ♠ët sè ❝❤➤♥ q t❛ ❝â x(1) = (b − p − 2) − q, x(3) = (b − p − 2) − q + 1, x(4) = (b − p − 2) − q + 3, ··· , x(k − p) = (b − p − 2) − ❳→❝ ✤✐♥❤ o ✈➔ o tø ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ (b − p − 2) − q = b − o − − o ✈➔ b − o − = b − q − 3✳ ❱➻ b✱ q ✱ ✈➔ p ❝❤➤♥ ♥❣❛② ❝↔ ❦❤✐ o ✈➔ o ❧➔ ❧➫✳ ❑➳t ❤ñ♣ ♥❤ú♥❣ ❦➳t q ợ t õ ỗ t tự ✸✸ ✭✐✐✐✮ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ r➡♥❣ ✭✷✳✶✸✮ ❧➔ ✤ó♥❣ ✈ỵ✐ p ≥ 1✱ p ❧➫✳ ✣➸ t❤✉➟♥ t✐➺♥ ❝❤♦ tr trữợ t t sỷ p ❑❤✐ ✤â = F−x(k) − Fb−2 + F−x(k−1) + + F−x(k−(p−1)) + F−x(k−p) + F−x(k−p−1) + + F−x(3) + Fx(1) , ( t❤❡♦ ✭✷✳✶✵✮✮ = Fb−1 − Fb−2 + (−Fb−2 + Fb−3 ) + (−Fb−4 + Fb−5 ) + + (−Fb−(p−1) + Fb−p ) + F−x(k−p) + F−x(k−p−1) + + F−x(3) + F−x(1) , t❤❡♦ ✭✷✳✶✸✮ = Fb−3 − Fb−4 − Fb−6 − − Fb−(p+1) + F−x(k−p) + F−x(k−p−1) + + F−x(3) + Fx(1) = Fb−(p+2) + F−x(k−p) + F−x(k−p−1) + + F−x(3) + Fx(1) , ✭✷✳✶✼✮ ❉♦ t➼♥❤ tè✐ ✤↕✐ ❝õ❛ p tr♦♥❣ ✭✷✳✶✸✮ t❛ ❝â x(j) : j ≤ k − p, x(j) ❝❤➤♥ ≤ b − p − 3, ✈ỵ✐ ≤ j ≤ p✳ ❉♦ ✤â →♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✸✳✻ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❝✉è✐ tr♦♥❣ ✭✷✳✶✼✮ t❛ ✤÷đ❝ Fb−p−2 + i∈J Fi = i∈K≤b−p−3 Fi < Fb−p−2 ◆➳✉ p = ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❚ø ❝→❝ ❦➳t q✉↔ tr➯♥ t õ ỵ s ỵ m ≥ ✈➔ p, x(1), x(3), x(4), , x(m) ❧➔ ♠ët ♥❣❤✐➺♠ ♥❣✉②➯♥ tè ❧ỵ♥✱ ❝❤➤♥ ❝õ❛ ữỡ tr sỷ r ợ ởt số i, x(i) b t tỗ t t j0 ✈ỵ✐ x(j0 ) = b✳ ✣➦t S1 = {x(i) : i < j0 } ✈➔ k = sup S1 t❤➻ ❤♦➦❝ k = ✈➔ x(k) = b−2 ❤♦➦❝ ♠ët tr♦♥❣ ✭✷✳✶✶✮✱ ✭✷✳✶✹✮✱ ✭✷✳✶✺✮ ♣❤↔✐ ✤ó♥❣✳ ✷✳✼ ❚r÷í♥❣ ❤đ♣ b r trữợ t ổ t ✤õ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ♥❣✉②➯♥ tè ❧ỵ♥✱ ❝❤➤♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✳ ❈❤ó♥❣ t❛ ✤➣ ❝❤➾ r❛ r➡♥❣ ❝→❝ ♥❣❤✐➺♠ ♥➔② ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ✾ ❞↕♥❣ tr♦♥❣ ✣à♥❤ ỵ ợ trữớ ủ b ❧➫ t❤➻ ✈➝♥ ❝❤÷❛ rã r➔♥❣✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❝→❝ ❜ê ✤➲ ❦❤➥♥❣ ✤à♥❤ r ổ õ tỗ t tr trữớ ủ b ❧➔ sè ❧➫✳ ✸✹ ❇ê ✤➲ ✷✳✼✳✶✳ ❑❤æ♥❣ ❝â ♥❣❤✐➺♠ tố ợ tr ữỡ tr ợ m > ✈➔ x(i) < b✱ ✈ỵ✐ ♠å✐ i✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ r➡♥❣ ❝â ♠ët ♥❣❤✐➺♠ ♥❣✉②➯♥ tố ợ tr b ữỡ tr ợ b m > ✈➔ x(i) < b✱ ✈ỵ✐ ♠å✐ i✳ ❚❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ x(1) ≤ b − ✈➔ x(i) : x(i) ❧➫ ≤ b − ❚❛ ①➨t ❜❛ tr÷í♥❣ ❤đ♣ s❛✉✿ ◆➳✉ x(1) = b − ✈➔ x(i) = b − ✈ỵ✐ ♠ët sè i ♥➔♦ ✤â✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❝♦♥ Fb = Fx(1) + F−x(i) ✈✐ ♣❤↕♠ ♥❣✉②➯♥ t➢❝ ❦❤✐ m > 3✳ ◆➳✉ x(1) = b − ✈➔ ❝→❝ ❝❤➾ sè x(i) ợ i ữủ tr ❜ð✐ b − ❙û ❞ö♥❣ ❇ê ✤➲ ✷✳✸✳✻✱ t❛ ❝â RST = Fx(1) + S (b − 2) < Fb−2 + Fb−1 = Fb ≤ LST, ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ◆➳✉ x(1) ≤ b − ✈➔ ❝→❝ số x(i) ợ i ữủ ❝❤➦♥ tr➯♥ ❜ð✐ b − 2✳ ❙û ❞ö♥❣ ❇ê ✤➲ ✷✳✸✳✻ t❛ ❝â RST = Fx(1) + S (b − 2) < Fb−2 + Fb−1 = Fb ≤ LST, ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â t❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✷✳✼✳✷✳ ❑❤æ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ tè ợ ữỡ tr ợ m > ✈➔ x(i) = b ✈ỵ✐ i ♥➔♦ ✤â✳ ❈❤ù♥❣ ♠✐♥❤✳ ❘ã r➔♥❣ ♥➳✉ b ❧➫ t❤➻ ♣❤÷ì♥❣ b tr➻♥❤ Fb = Fb ✈✐ ♣❤↕♠ t➼♥❤ ♥❣✉②➯♥ tè✳ ❉♦ ✤â t❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✷✳✼✳✸✳ ❑❤æ♥❣ ❝â tố ợ ữỡ tr ợ m > ✈➔ x(m) = b + o✳ ❈❤ù♥❣ sỷ ữủ õ tố ợ ữỡ tr b ợ b m > ✈➔ x(m) = b + o✳ ❱➻ b ❧➔ sè ❧➫ ♥➯♥ b + o ❧➔ sè ❝❤➤♥✳ ◆❤÷ ✈➟② ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❝→❝ ❝❤➾ sè ❧➫ ð ❜➯♥ ♣❤↔✐ ❧➔ b + o − 1✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ❇ê ✤➲ ✷✳✼✳✷ t❛ ❝â x(i) = b ✈ỵ✐ ♠å✐ i✳ ❙û ❞ư♥❣ ❇ê ✤➲ ✷✳✸✳✻ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ t❛ ❝â Fb + Fb+o + Fi < Fb−1 + Fb+o − Fb < Fb+o < Fb+o , Fi = Fx(1) + i∈J ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ i∈K≤b+o−1 rữợ t trữớ ủ ố ũ t ❜ê ✤➲ s❛✉✳ ❇ê ✤➲ ✷✳✼✳✹✳ ●✐↔ sû Fb = Fx(1) + F−x(3) + F−x(4) + + F−x(n) , ✈ỵ✐ < x(1) < n, < x(3) < x(4) < < x(n) ●✐↔ sû t❤➯♠ x(n), p ❧➔ ❝→❝ sè ❧➫ t❤ä❛ ♠➣♥ x(n) > p, n ≥ ❑❤✐ ✤â x (n − 1) = x(n) − ✈➔ x (n − 2) < x(n) − ự rữợ t ú t ự r x (n − 1) = x(n) − 1✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ r➡♥❣ x (n − 1) < x(n)−1✳ ⑩♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✸✳✻ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ♣❤→t ❜✐➸✉ ❝õ❛ ❜ê ✤➲ t❛ ❝â Fx(n) < Fx(1) +Fx(n) + Fi < Fp + i∈J Fi < Fp +Fx(n)−2 ≤ 2Fx(n)−2 < Fx(n) , i∈K≤x(n)−3 ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â x (n − 1) = x(n) − 1✳ ❚✐➳♣ t❤❡♦ t❛ ❝❤ù♥❣ ♠✐♥❤ x (n − 2) < x(n)−2✳ ●✐↔ sû ♥❣÷đ❝ ❧↕✐ x (n − 2) = x(n) − t❤➻ ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❝→❝ ❝❤➾ sè ❝❤➤♥ ð ❜➯♥ ♣❤↔✐ ❧➔ x(n) − 3✳ ❍ì♥ ♥ú❛✱ F−x(n) + F−x(n−1) + F−x(n−2) = 2Fx(n)−2 ✳ ❉♦ ✤â →♣ ❞ö♥❣ ❙û ❞ö♥❣ ❇ê ✤➲ ✷✳✸✳✻ t❛ ❝â 2Fx(n)−2 + Fx(1) + Fi < Fp + Fx(n)−2 ≤ 2Fx(n)−2 , Fi < Fp + i∈J i∈K≤x(n)−3 ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â x (n − 2) < x(n) − 2✳ ❇ê ✤➲ ✷✳✼✳✺✳ ❑❤æ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ tố ợ ữỡ tr ợ m > ✈➔ x(m) = b + o + 1✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x(m) = b + o + 1✳ rữợ t ợ b b = p, n = m✱ t❛ ❝â x (m − 1) = b + o ✈➔ x (m − 2) < b + o − 1✳ ❚❤❛② t❤➳ F−x(m) + F−x(m−1) = Fb+o+1 − Fb+o = Fb+o−1 ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ t❛ ❝â Fb = Fx(1) + F−x(3) + F−x(4) + + F−(b+o−1) ◆➳✉ (o − 1) ≥ 2✱ →♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✼✳✹ ✈ỵ✐ b = p, n = m − 1, t❛ ❝â x (m − 2) = b + o − ✈➔ x (m − 3) < b + o − 3✳ ❉♦ ✤â F−(b+o−1) + F−x(m−2) = F−(b+o−3) ✸✻ ❚✐➳♣ tö❝ q✉→ tr➻♥❤ ✤➳♥ ❦❤✐ ✤↕t ✤÷đ❝ ♠ët sè r s❛♦ ❝❤♦ x(m) = b + o + 1, x(m − 1) = b + o x(m − 2) = b + o − 2, · · · , x(m − r) = b + ❚❤❡♦ ❇ê ✤➲ ✷✳✸✳✹ t❛ ❝â F−x(m−r) + F−x(m−r+1) + + F−x(m−1) + F−x(m) = F−(b+1) + F−(b+3) + + F−(b+o) + F−(b+o+1) = Fb ❱➻ x(1) < b ♥➯♥ ❣✐↔ t❤✐➳t ❜❛♥ ✤➛✉ x(m) = b + o + ❧➔ ❦❤æ♥❣ ❝❤➼♥❤ ①→❝✳ ❚ø ❝→❝ ❇ê ✤➲ tr➯♥ t❛ ❝â ✤à♥❤ ỵ s ỵ ợ m > ổ õ tố ợ ữỡ tr ự ỵ ỵ ♥❤✐➯♥✮ ❈❤ù♥❣ ♠✐♥❤✳ ❛✮ ❱ỵ✐ m = 3✱ →♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✷✳✶ t❛ ❝â {b, x(1), x(3)} ❧➔ ♥❣❤✐➺♠ ợ ữỡ tr ợ x(1) = b 1, x(3) = b − 2✱ ❤♦➦❝ x(1) = b − 2, x(3) = b − 1✳ ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❜✮ ❱ỵ✐ m > ✲ ❍➺ q✉↔ ✷✳✸✳✺ ✤➣ ❦❤➥♥❣ ✤à♥❤ ✈ỵ✐ b ❧➔ sè ❝❤➤♥ ❜➜t ❦➻✱ ✈➔ ✈ỵ✐ o, o , o ❧➔ ❝→❝ số ữỡ t tr ỵ ✷✳✸✳✶ ✤➲✉ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✳ ✲ ◆➳✉ x(i) < b✱ ✈ỵ✐ ♠å✐ i✱ t❤➻ ❞↕♥❣ ✭✶✮ s➩ ữỡ tr t ỵ tỗ t i s x(i) b t ỵ t õ x(m) = b +1✱ ❤♦➦❝ x(m) = b + o +2, x(m − 1) = b + o +1, x(m − 2) = b + o − 1, , x(m − j) = b + ợ j ởt số ữỡ ♥➔♦ ✤â✳ ✲ ⑩♣ ❞ö♥❣ ❤➺ q✉↔ ✷✳✸✳✺✱ t❛ t❤➜② tỗ t j0 s x(j0 ) = b ✈ỵ✐ t➟♣ {x(i) : i > j0 } , ❤♦➦❝ t➟♣ {x(m) = b + 1} , ❤♦➦❝ ❧➔ t➟♣ {x(j0 + 1), x(j0 + 2), , x(m − 1), x(m)} ✸✼ = {b + 2, b + 4, , b + o + 1, b + o + 2} ❱➻ ✈➟② ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ s➩ ❝â ❝→❝ tứ tỗ t j0 ✤➸ x(j0 ) = b✱ ✈ỵ✐ t➟♣ {x(i) : i > j0 }✱ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✻✳✶✱ ❇ê ✤➲ ỵ t t ❝â ❞↕♥❣ ✭✷✳✶✶✮✱ ✭✷✳✶✹✮ ❤♦➦❝ ✭✷✳✶✺✮✳ ❚❛ s➩ ❝â ❤♦➦❝ ❧➔ t➟♣ {x(1) = {b − 1}} , ❤♦➦❝ ❧➔ {x(1), x(3), , x(j0 − 1)} = {b − o − 3, b − o − 1, b − o, , b − 1} , ❤♦➦❝ ❧➔ {x(1), x(3), , x(j0 − 1)} = {b − o − 3, b − o − 1, b − o, , b − 3} , ❤♦➦❝ ❧➔ {x(1), x(3), , x(j0 − 1)} ={b − o − − o , b − o − − o , b − o − − o , } ∪ {, , b − o − 4, b − o − 1, b − o, , b − 1} ❱➻ ✈➟② ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ s➩ ❧➔ ♠ët tr♦♥❣ ❝→❝ ❞↕♥❣ tø ✭✷✮ ✤➳♥ ✭✾✮✳ ✲ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ b ❧➔ sè ❧➫✱ ỵ t t ữỡ tr ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ♥❣✉②➯♥ tè ❧ỵ♥✳ ✸✽ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ♥❤÷ s❛✉✿ ✶✳ ❈æ♥❣ t❤ù❝ ❇✐♥❡t ❝❤♦ ❞➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ❞➣② ▲✉❝❛s✳ ✷✳ ❇➔✐ t♦→♥ ✼✼✾ ✈➔ ✽✵✹✳ ✸✳ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ tê♥❣ q✉→t ✈ỵ✐ ❞➣② ❋✐❜♦♥❛❝❝✐✳ ✹✳ ❚r➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ tr♦♥❣ tr÷í♥❣ ❤đ♣ m = 3, 4✳ ✺✳ ❚r➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✳ ✸✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆♥❞r❡✇ ❈✉s✉♠❛♥♦✳ Pr♦❜❧❡♠ ❇✲✼✼✾✳ ❚❤❡ ❋✐❜♦♥❛❝❝✐ ◗✉❛rt❡r❧② ✸✸✳✶ ✭✶✾✾✺✮ ✽✺✳ ❬✷❪ ▲✳ ❆✳ ● ❉r❡s❡❧✳ Pr♦❜❧❡♠ ❇✲✽✵✹✳ ❚❤❡ ❋✐❜♦♥❛❝❝✐ ◗✉❛rt❡r❧② ✸✺✳✶ ✭✶✾✾✼✮ ✽✽✳ ❬✸❪ ❚❤❡ Pr♦❜❧❡♠ ❊❞✐t♦r✳ Pr♦❜❧❡♠ ❇✲✼✼✾✳ ❚❤❡ ❋✐❜♦♥❛❝❝✐ ◗✉❛rt❡r❧② ✸✹✳✶ ✭✶✾✾✻✮ ✽✹✳ ❬✹❪ ❘✉❛✇r❧❧ ❏❛② ❍❡♥❞❡❧✳ ▲✐♥❡❛r ❡q✉❛❧✐t✐❡s ✐♥ ❋✐❜♦♥❛❝❝✐ ♥✉♠❜❡rs✳✭❡s❡♥t❡❞ ♦♥ t❤❡ ❖❝❝❛s❥♦♥ ♦❢ ❍❛r♦❧❞ ❙t❛r❦✬s ✻✺t❤ ❇✐rt❤❞❛②✮✳ ❙✉❜♠✐tt❡❞ ❆✉❣✉st ✷✵✵✹✲❋✐♥❛❧ ❘❡✈✐s✐♦♥ ❋❡❜r✉❛r② ✷✵✵✺✳ ... ❞➣② sè ▲✉❝❛s ✈➔ ❝→❝ ❞➣② t÷ì♥❣ tü✳ ●✐è♥❣ ♥❤÷ ❞➣② ❋✐❜♦♥❛❝❝✐✱ ♠é✐ sè tr♦♥❣ ❞➣② ▲✉❝❛s ❜➡♥❣ tờ số trữợ õ số ỗ t❤÷ì♥❣ ❣✐ú❛ ❤❛✐ sè ▲✉❝❛s ❧✐➲♥ ♥❤❛✉ s➩ ❤ë✐ tư ✤➳♥ ❣✐ỵ✐ ❤↕♥ ❜➡♥❣ t➾ ❧➺ ✈➔♥❣✳ ❚✉② ✈➟② ❦❤→❝... n, t❛ ❝â Ln = √ 1+ n − √ 1− n ❚ø ▼➺♥❤ ✤➲ t õ ỵ s ỵ t ố ỳ số ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ❞➣② ▲✉❝❛s✳ ✻ ỵ ợ số ữỡ n > m, t❛ ❝â FnLm = Fn+m + Fn−m ❱ỵ✐ ♠é✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ n t❛... s❛✉✳ ❚ø ♥❤➟♥ ①➨t 9342 = 9349 − = L19 − L4 , ð ✤➙② Lk ❧➔ sè ▲✉❝❛s t❤ù ❦✳ ❙û ❞ö♥❣ ❝→❝ ỗ t tự ỳ số số s t❛ ❝â Fm+19 − Fm−19 = Fm L19 , Fm+4 + Fm−4 = Fm L4 ❚rø ✈➳ ✈ỵ✐ ✈➳ ❝õ❛ ✷ ✤➥♥❣ t❤ù❝ tr➯♥ t❛