Open Propriedades e generalizações dos números de Fibonacci

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

Pr♦♣r✐❡❞❛❞❡s ❡ ●❡♥❡r❛❧✐③❛çõ❡s ❞♦s

◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐

♣♦r

❊❞❥❛♥❡ ●♦♠❡s ❞♦s ❙❛♥t♦s ❆❧♠❡✐❞❛

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❣♦st♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇

†_{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

❈❛t❛❧♦❣❛çã♦ ♥❛ ♣✉❜❧✐❝❛çã♦ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❇✐❜❧✐♦t❡❝❛ ❈❡♥tr❛❧

❆✹✹✼♣ ❆❧♠❡✐❞❛✱ ❊❞❥❛♥❡ ●♦♠❡s ❞♦s ❙❛♥t♦s✳

Pr♦♣r✐❡❞❛❞❡s ❡ ❣❡♥❡r❛❧✐③❛çõ❡s ❞♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ✴ ❊❞❥❛♥❡ ●♦♠❡s ❞♦s ❙❛♥t♦s✳✕ ❏♦ã♦ P❡ss♦❛✱ ✷✵✶✹✳

✹✸❢✳✿✐❧✳

❖r✐❡♥t❛❞♦r✿ ◆❛♣♦❧é♦♥ ❈❛r♦ ❚✉❡st❛ ❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮✲ ❯❋P❇✴❈❈❊◆✳

✶✳ ▼❛t❡♠át✐❝❛✳ ✷✳ ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳ ✸✳ ◆ú♠❡r♦s ❞❡ ▲✉❝❛s✳ ✹✳❋ór♠✉❧❛ ❞❡ ❇✐♥❡t✳ ✺✳❘❛③ã♦ ➪✉r❡❛✳ ✻✳◆ú♠❡r♦s ❚r✐❜♦♥❛❝❝✐✳

Pr♦♣r✐❡❞❛❞❡s ❡ ●❡♥❡r❛❧✐③❛çõ❡s ❞♦s

◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐

♣♦r

❊❞❥❛♥❡ ●♦♠❡s ❞♦s ❙❛♥t♦s ❆❧♠❡✐❞❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

ár❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛✳ ❆♣r♦✈❛❞❛ ♣♦r✿

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮

Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ ✲ ❯❋P❇

Pr♦❢✳ ❉r✳ ❚✉rí❜✐♦ ❏♦sé ●♦♠❡s ❞♦s ❙❛♥t♦s ✲ ❯◆■P✃

❆❣r❛❞❡❝✐♠❡♥t♦s

◗✉❡r♦ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s✱ ♣♦r ♠❡ ❣✉✐❛r ❡♠ t♦❞❛s ❛s ✈✐❛❣❡♥s ❞✉r❛♥t❡ ❡ss❡s ❞♦✐s ❛♥♦s ❡ ♠❡ ❞❛r ❢♦rç❛ ♣❛r❛ ✐r ❡♠ ❢r❡♥t❡✳

❆♦ Pr♦❢❡ss♦r ❉r✳ ◆❛♣♦❧❡♦♥ ❈❛r♦ ❚✉❡st❛✱ ❡①❡♠♣❧♦ ❞❡ ♣r♦✜ss✐♦♥❛❧ ❡ ❞❡ ❝♦♠♦ é ♣♦ssí✈❡❧ ❡①❡r❝❡r ❡ss❛ ♣r♦✜ssã♦ ♣♦✉❝♦ ✈❛❧♦r✐③❛❞❛ ❝♦♠ ❞✐❣♥✐❞❛❞❡✱ ❞❡❞✐❝❛çã♦ ❡ r❡s♣❡✐t♦ ♣❡❧♦ ❡st✉❞❛♥t❡✳ ❆♦ s❡♥❤♦r✱ ♠❡✉ r❡s♣❡✐t♦ ❡ ❛❞♠✐r❛çã♦✳

❆♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ❛♣♦✐♦✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ ♣❛✐✱ ♣♦r s❡♠♣r❡ ❛❝r❡❞✐t❛r ❡♠ ♠✐♥❤❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ✐r ❛❧é♠✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦✱ q✉❡ ♠✉✐t♦ ♠❡ ❛♣♦✐❛r❛♠ ❡ s❡♠ ♦s q✉❛✐s ❛ ❝♦♥❝❧✉sã♦ ❞♦ ❝✉rs♦ s❡r✐❛ ♠❛✐s ❞í✜❝✐❧✳ ▼❡✉ ♠✉✐t♦ ♦❜r✐❣❛❞❛ ♣❡❧❛ ❝♦❧❛❜♦r❛çã♦✳

❆♦ ♠❡✉ q✉❡r✐❞♦ ❡s♣♦s♦ ❏♦ã♦ ❘✐❝❛r❞♦✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❝♦♠♣r❡❡♥sã♦ ❡ ❞❡❞✐❝❛çã♦✳ ▼❡✉ ♠❛✐♦r ❛♣♦✐♦ ❞✉r❛♥t❡ t♦❞♦ ♦ ❝✉rs♦✳

❉❡❞✐❝❛tór✐❛

❘❡s✉♠♦

❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♦ ❡st✉❞♦ ❞♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳ ❆♣r❡s❡♥t❛✲s❡ ✐♥✐❝✐❛❧♠❡♥t❡ ✉♠ ❜r❡✈❡ r❡❧❛t♦ s♦❜r❡ ❛ ❤✐stór✐❛ ❞❡ ▲❡♦♥❛r❞♦ ❋✐❜♦♥❛❝❝✐✱ ❞❡s❞❡ s✉❛ ♦❜r❛ ♠❛✐s ❢❛♠♦s❛✱ ❖ ▲✐❜❡r ❆❜❛❝✐✱ ❛té ❛ r❡❧❛çã♦ ❝♦♠ ♦✉tr♦s ❝❛♠♣♦s ❞❛ ▼❛t❡♠át✐❝❛✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛✲s❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✱ ❛ ❋ór♠✉❧❛ ❞❡ ❇✐♥❡t✱ ♦s ◆ú♠❡r♦s ❞❡ ▲✉❝❛s ❡ ❛ r❡❧❛çã♦ ❝♦♠ ❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ ✉♠❛ ✐♠✲ ♣♦rt❛♥t❡ ♣r♦♣r✐❡❞❛❞❡ ♦❜s❡r✈❛❞❛ ♣♦r ❋❡r♠❛t✳ ❉❡♥tr♦ ❞❛s r❡❧❛çõ❡s ❝♦♠ ♦✉tr❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛✱ ❞❡st❛❝❛♠♦s ❛ r❡❧❛çã♦ ❝♦♠ ❛s ▼❛tr✐③❡s✱ ❝♦♠ ❛ ❚r✐❣♦♥♦♠❡tr✐❛✱ ❝♦♠ ❛ ●❡♦♠❡tr✐❛✳ ❆♣r❡s❡♥t❛✲s❡ t❛♠❜é♠ ❛ ❊❧✐♣s❡ ❡ ❛ ❍✐♣ér❜♦❧❡ ❞❡ ❖✉r♦✳ ❈♦♥❝❧✉✐♠♦s ❝♦♠ ♦s ◆ú♠❡r♦s ❚r✐❜♦♥❛❝❝✐ ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ r❡❣❡♠ ❡ss❡s ♥ú♠❡r♦s✳ ❘❡❛❧✐③❛✲ ♠♦s ❛❧❣✉♠❛s ❣❡♥❡r❛❧✐③❛çõ❡s s♦❜r❡ ▼❛tr✐③❡s ❡ P♦❧✐♥ô♠✐♦s ❚r✐❜♦♥❛❝❝✐✳

P❛❧❛✈r❛s✲❝❤❛✈❡ ▲❡♦♥❛r❞♦ ❋✐❜♦♥❛❝❝✐✱ ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✱ ◆ú♠❡r♦s ❞❡ ▲✉❝❛s✱ Pr♦♣r✐❡❞❛❞❡s✱ ❋ór♠✉❧❛ ❞❡ ❇✐♥❡t✱ ❘❛③ã♦ ➪✉r❡❛✱ ◆ú♠❡r♦s ❚r✐❜♦♥❛❝❝✐✳

❆❜str❛❝t

❚❤✐s ✇♦r❦ ✐s ❛❜♦✉t r❡s❡❛r❝❤ ❞♦♥❡ ❋✐❜♦♥❛❝❝✐✬s ◆✉♠❜❡rs✳ ■♥✐t✐❛❧❧② ✐t ♣r❡s❡♥ts ❛ ❜r✐❡❢ ❛❝❝♦✉♥t ♦❢ t❤❡ ❤✐st♦r② ♦❢ ▲❡♦♥❛r❞♦ ❋✐❜♦♥❛❝❝✐✱ ❢r♦♠ ❤✐s ♠♦st ❢❛♠♦✉s ✇♦r❦✱❚❤❡ ▲✐❜❡r ❆❜❛❝✐✱ t♦ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ✇✐t❤ ♦t❤❡r ✜❡❧❞s ♦❢ ▼❛t❤❡♠❛t✐❝s✳ ❚❤❡♥ ✇❡ ✇✐❧❧ ✐♥tr♦✲ ❞✉❝❡ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ ❋✐❜♦♥❛❝❝✐✬s ◆✉♠❜❡rs✱ ❇✐♥❡t✬s ❋♦r♠✱ ▲✉❝❛s✬ ◆✉♠❜❡rs ❛♥❞ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ✇✐t❤ ❋✐❜♦♥❛❝❝✐✬s ❙❡q✉❡♥❝❡ ❛♥❞ ❛♥ ✐♠♣♦rt❛♥t ♣r♦♣❡rt② ♦❜s❡r✈❡❞ ❜② ❋❡r♠❛t✳ ❲✐t❤✐♥ r❡❧❛t✐♦♥s❤✐♣s ✇✐t❤ ♦t❤❡r ❛r❡❛s ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✇❡ s❤♦✇ t❤❡ r❡❧❛✲ t✐♦♥s❤✐♣ ▼❛tr✐❝❡s✱ ❚r✐❣♦♥♦♠❡tr② ❛♥❞ ●❡♦♠❡tr②✳ ❆❧s♦ ♣r❡s❡♥ts t❤❡ ●♦❧❞❡♥ ❊❧❧✐♣s❡ ❛♥❞ t❤❡ ●♦❧❞❡♥ ❍②♣❡r❜♦❧❛✳ ❲❡ ❝♦♥❝❧✉❞❡ ✇✐t❤ ❚r✐❜♦♥❛❝❝✐✬s ◆✉♠❜❡rs ❛♥❞ s♦♠❡ ♣r♦✲ ♣❡rt✐❡s t❤❛t ❣♦✈❡r♥ t❤❡s❡ ♥✉♠❜❡rs✳ ▼❛❞❡ s♦♠❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s ❛❜♦✉t ▼❛tr✐❝❡s ❛♥❞ P♦❧②♥♦♠✐❛❧s ❚r✐❜♦♥❛❝❝✐✳

❑❡②✇♦r❞s✿ ▲❡♦♥❛r❞♦ ❋✐♥♦♥❛❝❝✐✱ ❋✐❜♦♥❛❝❝✐✬s ◆✉♠❜❡rs✱ ▲✉❝❛s✬ ◆✉♠❜❡rs✱ Pr♦♣❡r✲ t✐❡s✱ ❇✐♥❡t✬s ❋♦r♠✱ ●♦❧❞❡♥ ❘❛t✐♦✱ ❚r✐❜♦♥❛❝❝✐✬s ◆✉♠❜❡rs✳

❙✉♠ár✐♦

✶ ❍✐stór✐❛ ❡ ❖❜r❛ ❞❡ ▲❡♦♥❛r❞♦ ❋✐❜♦♥❛❝❝✐ ✶

✶✳✶ ❍✐stór✐❛ ❡ ❖❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ ❖ ▲✐❜❡r ❆❜❛❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✷ ❖ Pr♦❜❧❡♠❛ ❞❛ ❘❡♣r♦❞✉çã♦ ❞♦s ❈♦❡❧❤♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷

✷ Pr♦♣r✐❡❞❛❞❡s ❞♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ✹

✷✳✶ ❉❡✜♥✐çã♦ ❞❡ ❙❡q✉ê♥❝✐❛ ❘❡❝✉rs✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✷ Pr♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✸ ❋ór♠✉❧❛ ❞❡ ❇✐♥❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✹ ❋✐❜♦♥❛❝❝✐ ❡ ❛ ❙❡q✉ê♥❝✐❛ ❞❡ ▲✉❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✺ ❋❡r♠❛t ❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸ ❘❡❧❛çã♦ ❞♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❝♦♠ ♦✉tr❛s ár❡❛s ❞❛ ▼❛t❡♠á✲

t✐❝❛ ✶✾

✸✳✶ ❋✐❜♦♥❛❝❝✐ ❡ ❛ ❚r✐❣♦♥♦♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✷ ❋✐❜♦♥❛❝❝✐ ❡ ❛s ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✸ ❋✐❜♦♥❛❝❝✐ ❡ ❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✸✳✶ ❖ ◆ú♠❡r♦ ❞❡ ❖✉r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✸✳✷ ❖ ❘❡tâ♥❣✉❧♦ ➪✉r❡♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✸✳✸ ❉✐✈✐sã♦ ➪✉r❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✸✳✹ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✹ ❆ ❊❧✐♣s❡ ❡ ❛ ❍✐♣ér❜♦❧❡ ❞❡ ❖✉r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✹✳✶ ❆ ❊❧✐♣s❡ ❞❡ ♦✉r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✹✳✷ ❆ ❍✐♣ér❜♦❧❡ ❞❡ ❖✉r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✺ ◆ú♠❡r♦s ❚r✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✺✳✶ ❖s ◆ú♠❡r♦s ❚r✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✺✳✷ ▼❛tr✐③❡s ❚r✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✺✳✸ ❈♦♠♣♦♥❞♦ ❙♦♠❛s ❝♦♠ ✶✱ ✷ ❡ ✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✺✳✹ ❋✉♥çã♦ ●❡r❛❞♦r❛ ♣❛r❛ Tn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✸✳✺✳✺ P♦❧✐♥ô♠✐♦s ❚r✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✶

■♥tr♦❞✉çã♦

▲❡♦♥❛r❞♦ ❞❡ P✐s❛ ♦✉ ▲❡♦♥❛r❞♦ ❋✐❜♦♥❛❝❝✐ ♥❛s❝❡✉ ❡♠ P✐s❛✱ ♥❛ ■tá❧✐❛✳ ❆❝♦♠♣❛✲ ♥❤❛♥❞♦ s✉❛ ❢❛♠í❧✐❛✱ ♠✉✐t♦ ❝❡❞♦ ❢♦✐ ♣❛r❛ ❛ ❆r❣é❧✐❛ ❡ ❧á✱ r❡❝❡❜❡✉ s✉❛ ❡❞✉❝❛çã♦ ❝♦♠ ♣r♦❢❡ss♦r❡s ♠✉ç✉❧♠❛♥♦s✱ t❡♥❞♦ ❝♦♥t❛t♦ ❝♦♠ ❛ ♠❛t❡♠át✐❝❛ ✐♥❞♦✲❛rá❜✐❝❛✳ ❋❡③ ✈ár✐❛s ✈✐❛❣❡♥s ♣❡❧♦ ♠❡❞✐t❡rrâ♥❡♦ ❡ ✜❝♦✉ ❝♦♥✈❡♥❝✐❞♦ ❞❛ s✉♣❡r✐♦r✐❞❛❞❡ ❞♦ s✐st❡♠❛ ❞❡❝✐♠❛❧ ✐♥❞♦✲❛rá❜✐❝♦✳ ❆♦ ✈♦❧t❛r ♣❛r❛ ❛ ■tá❧✐❛✱ ❡s❝r❡✈❡✉ ♦ ▲í❜❡r ❆❜❛❝✐✳

❖ ▲í❜❡r ❆❜❛❝✐ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❧✐✈r♦ ❡s❝r✐t♦ ♣♦r ❋✐❜♦♥❛❝❝✐✱ ♥❡❧❡✱ ❋✐❜♦♥❛❝❝✐ ❛♣r❡s❡♥✲ t♦✉ ❊✉r♦♣❛ ♦ s✐st❡♠❛ ❞❡ ♥✉♠❡r❛çã♦ ✐♥❞♦✲ár❛❜❡ ❡ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❝♦♠ ❝♦♥✈❡rsõ❡s ♠♦♥❡tár✐❛s✱ ❝á❧❝✉❧♦ ❞❡ ❥✉r♦s ❡ ♦ s❡✉ ♠❛✐s ❢❛♠♦s♦✱ ♦ Pr♦❜❧❡♠❛ ❞❛ ❘❡♣r♦❞✉çã♦ ❞♦s ❈♦❡❧❤♦s✳ ❆ s♦❧✉çã♦ ❞❡ss❡ ♣r♦❜❧❡♠❛ é ❡①❛t❛♠❡♥t❡ ❛ ❝♦♥❤❡❝✐❞❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦✲ ♥❛❝❝✐✳

❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❣❡♥❡r❛❧✐③❛çõ❡s ❞♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦✲ ♥❛❝❝✐✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞❡ss❡ ♥♦tór✐♦ ♠❛t❡♠át✐❝♦✱ s✉❛ ♦❜r❛ ♠❛✐s ✐♠♣♦rt❛♥t❡✿ ❖ ▲✐❜❡r ❆❜❛❝✐ ❡ ❛ ❝♦♥tr✐❜✉✐çã♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❚❡♦t✐❛ ❞♦s ◆ú♠❡r♦s✳

❘❡❛❧✐③❛✲s❡ t❛♠❜é♠ ✉♠ ✐♠♣♦rt❛♥t❡ ❡st✉❞♦ s♦❜r❡ ❛ r❡❧❛çã♦ ❞♦s ◆ú♠❡r♦s ❞❡ ❋✐✲ ❜♦♥❛❝❝✐ ❡ ♦s ◆ú♠❡r♦s ❞❡ ▲✉❝❛s✱ ❛s ♣r✐♥❝✐♣❛✐s ❣❡♥❡r❛❧✐③❛çõ❡s ❡ r❡❧❛çõ❡s ❝♦♠ ♦✉tr❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛✳ ❉❡st❛❝❛♥❞♦ ❛ r❡❧❛çã♦ ❝♦♠ ❛ tr✐❣♦♥♦♠❡tr✐❛✱ ❝♦♠ ❛ ❣❡♦♠❡tr✐❛ ❡ ❝♦♠ ❛s ♠❛tr✐③❡s✳

❚r❛③❡♠♦s ❛q✉✐ ✉♠❛ ❜r❡✈❡ ❛♣r❡s❡♥t❛çã♦ ❞♦s ♥ú♠❡r♦s ❚r✐❜♦♥❛❝❝✐✱ ♣r♦✈❛♠♦s ❛❧❣✉✲ ♠❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❣❡♥❡r❛❧✐③❛♠♦s ❛s ▼❛tr✐③❡s ❡ ♦s P♦❧✐♥ô♠✐♦s ❚r✐❜♦♥❛❝❝✐✳

❈❛♣ít✉❧♦ ✶

❍✐stór✐❛ ❡ ❖❜r❛ ❞❡ ▲❡♦♥❛r❞♦

❋✐❜♦♥❛❝❝✐

❊st✉❞❛r❡♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ ❛ ❤✐stór✐❛ ❡ ❛ ♦❜r❛ ❞❡ ▲❡♦♥❛r❞♦ ❋✐❜♦♥❛❝❝✐✳ ❘❡ss❛❧✲ t❛r❡♠♦s s✉❛ ♦❜r❛ ♠❛✐s ❝♦♥❤❡❝✐❞❛✱ ♦ ▲✐❜❡r ❆❜❛❝✐ ❡ ♦ Pr♦❜❧❡♠❛ ❞❛ ❘❡♣r♦❞✉çã♦ ❞♦s ❈♦❡❧❤♦s✱ ❞❡st❛❝❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✶✷ ❞♦ ▲✐❜❡r ❆❜❛❝✐ ❡ q✉❡ ❞❡✉ ♦r✐❣❡♠ ❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ♦✉ ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳

✶✳✶ ❍✐stór✐❛ ❡ ❖❜r❛

▲❡♦♥❛r❞♦ ❞❡ P✐s❛ ♦✉ ▲❡♦♥❛r❞♦ P✐s❛♥♦✱ ♥❛s❝❡✉ ❡♠ P✐s❛ ♥❛ ❚♦s❝â♥✐❛ ✭■tá❧✐❛✮ ♣♦r ✈♦❧t❛ ❞❡ ✶✳✶✼✵✳ ❋✐❝♦✉ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ▲❡♦♥❛r❞♦ ❋✐❜♦♥❛❝❝✐✱ ❋✐❜♦♥❛❝❝✐ s✐❣♥✐✜❝❛ ✜❧❤♦ ❞❡ ❇♦♥❛❝❝✐♦✳ ❙❡✉ ♣❛✐ ●✉❣❧✐❡❧♠♦ ❇♦♥❛❝❝✐ ❡r❛ ✉♠ ❣r❛♥❞❡ ♠❡r❝❛❞♦r ❡ ❢♦r❛ ♥♦♠❡❛❞♦ ❝♦❧❡t♦r ❞❛s ❆❧❢â♥❞❡❣❛s ♥❛ ❆r❣é❧✐❛✱ ❝✐❞❛❞❡ ❞❡ ❇✉❣✐❛ ✭❛❣♦r❛ ❇♦✉❣✐❡✮✳ ❊❧❡ ❧❡✈♦✉ ▲❡♦✲ ♥❛r❞♦ ♣❛r❛ ❛♣r❡♥❞❡r ❛ ❛rt❡ ❞❡ ❝❛❧❝✉❧❛r✳

❊♠ ❇♦✉❣✐❡✱ ▲❡♦♥❛r❞♦ r❡❝❡❜❡✉ s✉❛ ❡❞✉❝❛çã♦ ❞❡ ✉♠ ♣r♦❢❡ss♦r ♠✉ç✉❧♠❛♥♦✱ q✉❡ ♦ ❛♣r❡s❡♥t♦✉ ❛♦ s✐st❡♠❛ ❞❡ ♥✉♠❛r❛çã♦ ❡ ❛s té❝♥✐❝❛s ❞❡ ❝á❧❝✉❧♦ ✐♥❞♦✲❛rá❜✐❝♦s✳ ❋♦✐ ❛í t❛♠❜é♠ q✉❡ ❋✐❜♦♥❛❝❝✐ t❡✈❡ ❛❝❡ss♦ ❛♦ ❧✐✈r♦ ❞❡ á❧❣❡❜r❛ ❞♦ ♠❛t❡♠át✐❝♦ ♣❡rs❛ ❛❧✲ ❦❤♦✇❛r✐③♠✐✳

❏á ❛❞✉❧t♦✱ ❋✐❜♦♥❛❝❝✐ ❢❡③ ✈ár✐❛s ✈✐❛❣❡♥s ♣❡❧♦ ❊❣✐t♦✱ ❙ír✐❛✱ ●ré❝✐❛✱ ❋r❛♥ç❛ ❡ ❈♦♥s✲ t❛♥t✐♥♦♣❧❛✱ ♦♥❞❡ ❡st✉❞♦✉ ❞✐✈❡rs♦s s✐st❡♠❛s ❞❡ ♥✉♠❡r❛çã♦✳ P♦r ✈♦❧t❛ ❞❡ ✶✷✵✵✱ ✈♦❧t♦✉ ♣❛r❛ P✐s❛✳ ❈♦♥✈❡♥❝✐❞♦ ❞❛ s✉♣❡r✐♦r✐❞❛❞❡ ❡ ♣r❛t✐❝✐❞❛❞❡ ❞♦ ❙✐st❡♠❛ ❞❡ ◆✉♠❡r❛çã♦ ✐♥❞♦✲ár❛❜❡✱ ❡♠ ✶✷✵✷✱ ♣✉❜❧✐❝❛ s❡✉ ♣r✐♠❡✐r♦ tr❛❜❛❧❤♦✱ ♦ ▲✐❜❡r ❆❜❛❝✐ ✭❖ ❧✐✈r♦ ❞♦ ❈á❧✲ ❝✉❧♦✮✳

❋✐❜♦♥❛❝❝✐ t❛♠❜é♠ ❡s❝r❡✈❡✉ três ♦✉tr♦s ❧✐✈r♦s ✐♠♣♦rt❛♥t❡s✳ Pr❛❝t✐❝❛ ❞❡ ●❡♦♠❡✲ tr✐❛❡ ✭Prát✐❝❛ ❞❛ ❣❡♦♠❡tr✐❛✮✱ ❡s❝r✐t♦ ❡♠ ✶✷✷✵✱ ❛♣r❡s❡♥t❛ ❣❡♦♠❡tr✐❛ ❡ tr✐❣♦♥♦♠❡tr✐❛

❍✐stór✐❛ ❡ ❖❜r❛ ❈❛♣ít✉❧♦ ✶

❝♦♠ r✐❣♦r ❡✉❝❧✐❞✐❛♥♦ ❛ ❛❧❣✉♠❛ ♦r✐❣✐♥❛❧✐❞❛❞❡✳ ◆❡ss❡ ❧✐✈r♦✱ ❋✐❜♦♥❛❝❝✐ ❡♠♣r❡❣❛ á❧❣❡❜r❛ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s ❡ ❣❡♦♠❡tr✐❛ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❛❧❣é❜r✐❝♦s✱ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❛✈❛♥ç❛❞❛ ♣❛r❛ ❛ ❊✉r♦♣❛ ❞❡ s✉❛ é♣♦❝❛✳

❋❧♦s✱ ❡s❝r✐t♦ ❡♠ ✶✷✷✺✱ ❛♣r❡s❡♥t❛ ❛ s♦❧✉çã♦ ❞❡ três ♣r♦❜❧❡♠❛s q✉❡ ❢♦r❛♠ ❝♦❧♦❝❛❞♦s ♣❛r❛ ❋✐❜♦♥❛❝❝✐ ♣♦r ❏♦ã♦ ❞❡ P❛❧❡r♠♦✱ ✉♠ ♠❡♠❜r♦ ❞❛ ❈♦rt❡ ❞♦ ■♠♣❡r❛❞♦r ❋r❡❞❡r✐❝♦ ■■✳

▲✐❜❡r ◗✉❛❞r❛t♦r✉♠✱ t❛♠❜é♠ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✷✷✺✱ é ❝♦♥s✐❞❡r❛❞♦ ♦ ♠❛✐♦r ❧✐✈r♦ q✉❡ ❋✐❜♦♥❛❝❝✐ ❡s❝r❡✈❡✉✱ ♥♦ q✉❛❧ ❛♣r♦①✐♠❛ r❛í③❡s ❝ú❜✐❝❛s✱ ♦❜t❡♥❞♦ r❡s✉❧t❛❞♦s ❝♦rr❡t♦s ❛té ❛ ♥♦♥❛ ❝❛s❛ ❞❡❝✐♠❛❧✳

✶✳✶✳✶ ❖ ▲✐❜❡r ❆❜❛❝✐

❋♦✐ ❡s❝r✐t♦ ♣♦r ❋✐❜♦♥❛❝❝✐ ❡♠ ✶✷✵✷✱ ❜❛s❡❛❞♦ ❡♠ s❡✉s ❡st✉❞♦s r❡❛❧✐③❛❞♦s ♥♦ ♣❡rí♦❞♦ ❞❛s ✈✐❛❣❡♥s ♣❡❧♦ ▼❡❞✐t❡rrâ♥❡♦✳ ❆♣ós r❡❛❧✐③❛r ✉♠❛ r❡✈✐sã♦✱ ♣✉❜❧✐❝♦✉✲♦ ♥♦✈❛♠❡♥t❡ ❡♠ ✶✷✷✽✳

❖r❣❛♥✐③❛❞♦ ❡♠ ✶✺ ❝❛♣ít✉❧♦s✱ ♦ ❧✐✈r♦ t❡♠ ✉♠❛ ❢♦rt❡ ✐♥✢✉ê♥❝✐❛ ár❛❜❡✱ ❛♣r❡s❡♥t❛ ❛ ❧❡✐t✉r❛ ❡ ❛ ❡s❝r✐t❛ ❞♦s ♥ú♠❡r♦s ♥♦ s✐st❡♠❛ ❞❡❝✐♠❛❧ ✐♥❞♦✲ár❛❜❡✱ tr❛③ r❡❣r❛s ❞❡ ❝á❧❝✉❧♦✱ ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s q✉❡ ✐♥❝❧✉❡♠ q✉❡stõ❡s ❞❡ ❝á❧❝✉❧♦ ❞❡ ❥✉r♦s✱ ❝♦♥✈❡rsõ❡s ♠♦♥❡tár✐❛s ❡ ♠❡❞✐❞❛s✳ ❍á ✉♠❛ ❣r❛♥❞❡ ❝♦❧❡çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ❞❡♥tr❡ ♦s q✉❛✐s ♦ q✉❡ ❞❡✉ ♦r✐✲ ❣❡♠ à s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✿ ❖ Pr♦❜❧❡♠❛ ❞❛ ❘❡♣r♦❞✉çã♦ ❞♦s ❈♦❡❧❤♦s✱ t❛♠❜é♠ ❝♦♥s✐❞❡r❛❞♦ ♦ ♠❛✐s ❢❛♠♦s♦ ❞♦s ♣r♦❜❧❡♠❛s ❞❡ ▲❡♦♥❛r❞♦✳ ❆♣r❡s❡♥t❛ t❛♠❜é♠✱ r❛í③❡s q✉❛❞r❛❞❛s ❡ r❛í③❡s ❝ú❜✐❝❛s✳

✶✳✶✳✷ ❖ Pr♦❜❧❡♠❛ ❞❛ ❘❡♣r♦❞✉çã♦ ❞♦s ❈♦❡❧❤♦s

◆♦ ❈❛♣ít✉❧♦ ✶✷ ❞♦ ▲í❜❡r ❆❜❛❝✐✱ ▲❡♦♥❛r❞♦ ❛♣r❡s❡♥t❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ r❡♣r♦❞✉çã♦ ❞♦s ❈♦❡❧❤♦s✳ ❖ q✉❛❧ tr❛③ ✉♠❛ s✐t✉❛çã♦ ❤✐♣♦tét✐❝❛ ❞❡s❝r✐t❛ ❛ s❡❣✉✐r✿

❯♠❛ ♣❡ss♦❛ t❡♠ ✉♠ ♣❛r ❞❡ ❝♦❡❧❤♦s r❡❝é♠ ♥❛s❝✐❞♦s✱ ♥✉♠ ❧✉❣❛r ❝❡r❝❛❞♦ ♣♦r t♦❞♦s ♦s ❧❛❞♦s ♣♦r ✉♠ ♠✉r♦✳ ◗✉❛♥t♦s ♣❛r❡s ❞❡ ❝♦❡❧❤♦s ♣♦❞❡♠ s❡r ❣❡r❛❞♦s ❛ ♣❛rt✐r ❞❡ss❡ ♣❛r ❡♠ ✉♠ ❛♥♦ s❡✱ s✉♣♦st❛♠❡♥t❡✱ t♦❞♦ ♠ês ❝❛❞❛ ♣❛r ❞á ❛ ❧✉③ ❛ ✉♠ ♥♦✈♦ ♣❛r✱ q✉❡ é ❢ért✐❧ ❛ ♣❛rt✐r ❞♦ s❡❣✉♥❞♦ ♠ês✳

❊ss❡ ♣r♦❜❧❡♠❛✱ ❛♣❛r❡♥t❡♠❡♥t❡ ❞❡ s♦❧✉çã♦ s✐♠♣❧❡s✱ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❛ ✉♠❛ ❞❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❡s❝♦❜❡rt❛s ❞❛ ♠❛t❡♠át✐❝❛✳

■♥✐❝✐❛♠♦s ❝♦♠ ✉♠ ♣❛r ❥♦✈❡♠✱ ❛♣ós ♦ ♣r✐♠❡✐r♦ ♠ês✱ ❡ss❡ ♣❛r ❥á ❡stá ❛❞✉❧t♦ ❡ ❢ért✐❧✳ ◆♦ s❡❣✉♥❞♦ ♠ês✱ ❡ss❡ ♣r✐♠❡✐r♦ ♣❛r ❞á ❛ ❧✉③ ❛ ✉♠ ♦✉tr♦✱ ✜❝❛♥❞♦ ❝♦♠ ✷ ♣❛r❡s✳

❍✐stór✐❛ ❡ ❖❜r❛ ❈❛♣ít✉❧♦ ✶

◆♦ t❡r❝❡✐r♦ ♠ês✱ ♦ ♣❛r ❛❞✉❧t♦ ❞á ❛ ❧✉③ ❛ ♦✉tr♦ ♣❛r ❥♦✈❡♠✱ ❡♥q✉❛♥t♦ ♦ ♣❛r ❞❡ ✜❧❤♦t❡s t♦r♥❛✲s❡ ❢ért✐❧✱ ✜❝❛♥❞♦ ❛❣♦r❛ ❝♦♠ ✸ ♣❛r❡s✳

◆♦ q✉❛rt♦ ♠ês ❝❛❞❛ ✉♠ ❞♦s ❞♦✐s ♣❛r❡s ❛❞✉❧t♦s ❞á ❛ ❧✉③ ❛ ✉♠ ♣❛r ❥♦✈❡♠ ❡ ♦ t❡r❝❡✐r♦ ♣❛r t♦r♥❛✲s❡ ❛❞✉❧t♦ ❡ ❢ért✐❧✳

❆ t❛❜❡❧❛ ❛ s❡❣✉✐r ♠♦str❛ ❛ r❡♣r♦❞✉çã♦ ❞♦s ❝♦❡❧❤♦s ❛té ♦ ❞é❝✐♠♦ s❡❣✉♥❞♦ ♠ês✳

◆ú♠❡r♦ ❞❡

♣❛r❡s ❞❡ ❝♦✲

❡❧❤♦s r❡❝é♠

♥❛s❝✐❞♦s

◆ú♠❡r♦ ❞❡

♣❛r❡s ❞❡ ❝♦✲

❡❧❤♦s ❛❞✉❧✲

t♦s

◆ú♠❡r♦ t♦✲

t❛❧ ❞❡ ♣❛r❡s

❞❡ ❝♦❡❧❤♦s

■♥í❝✐♦

✶

✵

✶

❯♠ ♠ês ❞❡♣♦✐s

✵

✶

✶

❉♦✐s ♠❡s❡s ❞❡♣♦✐s

✶

✶

✷

❚rês ♠❡s❡s ❞❡♣♦✐s

✶

✷

✸

◗✉❛tr♦ ♠❡s❡s ❞❡♣♦✐s ✷

✸

✺

❈✐♥❝♦ ♠❡s❡s ❞❡♣♦✐s ✸

✺

✽

❙❡✐s ♠❡s❡s ❞❡♣♦✐s

✺

✽

✶✸

❙❡t❡ ♠❡s❡s ❞❡♣♦✐s

✽

✶✸

✷✶

❖✐t♦ ♠❡s❡s ❞❡♣♦✐s

✶✸

✷✶

✸✹

◆♦✈❡ ♠❡s❡s ❞❡♣♦✐s ✷✶

✸✹

✺✺

❉❡③ ♠❡s❡s ❞❡♣♦✐s

✸✹

✺✺

✽✾

❖♥③❡ ♠❡s❡s ❞❡♣♦✐s ✺✺

✽✾

✶✹✹

❉♦③❡ ♠❡s❡s ❞❡♣♦✐s ✽✾

✶✹✹

✷✸✸

❆ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ♥♦s ❞á ❛ s❡q✉ê♥❝✐❛✿ ✶✱ ✶✱ ✷✱ ✸✱ ✺✱ ✽✱ ✶✸✱✷✶✱ ✸✹✱ ✺✺✱✽✾✱✶✹✹✱✷✸✸✳

P♦❞❡♠♦s ❡①❛♠✐♥❛r s♦♠❡♥t❡ ♦ ♥ú♠❡r♦ ❞❡ ♣❛r❡s ❞❡ ❝♦❡❧❤♦s ❛❞✉❧t♦s ❡♠ ✉♠ ❞❡t❡r✲ ♠✐♥❛❞♦ ♠ês✱ ❡ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❡ss❡ ♥ú♠❡r♦ é ❢♦r♠❛❞♦ ♣❡❧❛ s♦♠❛ ❞♦s ♣❛r❡s ❛❞✉❧t♦s ❞♦s ✷ ♠❡s❡s ❛♥t❡r✐♦r❡s✱ ❡ ❛ ♠❡s♠❛ ❡①♣❡r✐ê♥❝✐❛ ✈❛❧❡ ♣❛r❛ ♦s ♣❛r❡s ❥♦✈❡♥s✳

◆♦ sé❝✉❧♦ ❳■❳ ❡ss❛ s❡q✉ê♥❝✐❛ ❢♦✐ ❝❤❛♠❛❞❛ ❞❡ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ♣❡❧♦ ♠❛✲ t❡♠át✐❝♦ ❢r❛♥❝ês ❊❞♦✉❛r❞ ▲✉❝❛s ✭✶✽✹✷✲✶✽✾✶✮✳

❈❛♣ít✉❧♦ ✷

Pr♦♣r✐❡❞❛❞❡s ❞♦s ◆ú♠❡r♦s ❞❡

❋✐❜♦♥❛❝❝✐

❊st✉❞❛r❡♠♦s ❛❣♦r❛ ❛ ❞❡✜♥✐çã♦ ❞❡ s❡q✉ê♥❝✐❛s r❡❝✉rs✐✈❛s ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛✲ ❞❡s ❞♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳ ❊st❡ ❝❛♣ít✉❧♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❡♠ ❝✐♥❝♦ s❡çõ❡s✳ ❆ ♣r✐♠❡✐r❛ ❛♣r❡s❡♥t❛ ❛ ❞❡✜♥✐çã♦ ❞❡ s❡q✉ê♥❝✐❛s r❡❝✉rs✐✈❛s❀ ♥❛ s❡❣✉♥❞❛✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s r❡❧❛❝✐♦♥❛❞❛s ❛♦ ♠❞❝ ❡ ❛ s♦♠❛ ❞❡ ♥ú♠❡r♦s ❞❛ s❡q✉ê♥❝✐❛✳ ◆❛ t❡r❝❡✐r❛✱ ❛ ❋ór♠✉❧❛ ❞❡ ❇✐♥❡t❀ ♥❛ q✉❛rt❛✱ ❛ r❡❧❛çã♦ ❝♦♠ ❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ ▲✉❝❛s ❡ ✜♥❛❧✐③❛♥❞♦ ♦ ❝❛♣ít✉❧♦✱ ♥❛ q✉✐♥t❛ s❡çã♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ❢❛s❝✐♥❛♥t❡ ♣r♦♣r✐❡❞❛❞❡ ♦❜s❡r✈❛❞❛ ♣♦r ❋❡r♠❛t✳

✷✳✶ ❉❡✜♥✐çã♦ ❞❡ ❙❡q✉ê♥❝✐❛ ❘❡❝✉rs✐✈❛

❆♦ ♦❜s❡r✈❛r♠♦s ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ✈❡r✐✜❝❛♠♦s q✉❡ ❝❛❞❛ t❡r♠♦✱ ❛ ♣❛rt✐r ❞♦ t❡r❝❡✐r♦ é ✐❣✉❛❧ ❛ s♦♠❛ ❞❡ ❞♦✐s t❡r♠♦s ❛♥t❡r✐♦r❡s✳ ❆s s❡q✉ê♥❝✐❛s q✉❡ sã♦ ❞❡✜♥✐❞❛s ❞❡ss❛ ❢♦r♠❛ sã♦ ❝❤❛♠❛❞❛s ❞❡ ❙❡q✉ê♥❝✐❛s ❘❡❝✉rs✐✈❛s✳

❙❛❜❡♥❞♦ ❞❡ss❛ ♣r♦♣r✐❡❞❛❞❡✱ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❝♦♥s✐❞❡✲ r❛♥❞♦✿

✶✳ F0 = 0, F1 = 1 ✭❈♦♥❞✐çã♦ ✐♥✐❝✐❛❧✮

✷✳ Fn=Fn−1+Fn−2, n ≥2✭❘❡❧❛çã♦ ❞❡ ❘❡❝♦rrê♥❝✐❛✮ ❱❡❥❛♠♦s✿

❋2 =F1+F0 = 1 + 0 = 1 ❋3 =F2+F1 = 1 + 1 = 2 ❋4 =F3+F2 = 2 + 1 = 3 ❋5 =F4+F3 = 3 + 2 = 5 ❋6 =F5+F4 = 5 + 3 = 8

Pr♦♣r✐❡❞❛❞❡s ❈❛♣ít✉❧♦ ✷

❉❛í✱ ♦❜t❡♠♦s ❛ s❡q✉ê♥❝✐❛✿ 0,1,1,2,3,5,8, ...

❆ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❢♦✐ ✉♠❛ ❞❛s ♣r✐♠❡✐r❛s s❡q✉ê♥❝✐❛s r❡❝✉rs✐✈❛s ❝♦♥❤❡❝✐❞❛ ♥❛ ❊✉r♦♣❛✳

✷✳✷ Pr♦♣r✐❡❞❛❞❡s

❖s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❛♣r❡s❡♥t❛♠ ✈ár✐❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ♠✉✐t❛s ❞❡❧❛s ❢♦r❛♠ ❡st✉❞❛❞❛s ♣♦r ✈ár✐♦s ♠❛t❡♠át✐❝♦s ❛♦ ❧♦♥❣♦ ❞♦s ❛♥♦s✳ ❆q✉✐ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❞❡❧❛s✳

❆♦ s❡ ♦❜s❡r✈❛r ❞♦✐s t❡r♠♦s ❝♦♥s❡❝✉t✐✈♦s ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ✈❡r✐✜❝❛✲s❡ q✉❡ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ ❡❧❡s é ✐❣✉❛❧ ❛ ✶✳ ❈♦♥s✐❞❡r❡ ♦ F6 = 8 ❡ ♦ F7 = 13 ❡ ♦

(8,13) = 1✳ ■ss♦ ♣♦rq✉❡ ♦s ❞✐✈✐s♦r❡s ♣♦s✐t✐✈♦s ❞❡ F6 = 8 sã♦ 1,2,4,8 ❡ ♦s ❞✐✈✐s♦r❡s ♣♦s✐t✐✈♦s ❞❡ F7 = 13 sã♦ 1,13✳ ❖❜s❡r✈❛♥❞♦ ♦✉tr♦s t❡r♠♦s ❝♦♥s❡❝✉t✐✈♦s✱ ♣❡r❝❡❜❡♠♦s q✉❡ ❤á ✉♠❛ r❡❣✉❧❛r✐❞❛❞❡✳ ❊ ❞❛í ♣♦❞❡♠♦s ❡♥✉♥❝✐❛r ❛ ♣r✐♠❡✐r❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦s ♥ú✲ ♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳ ❉❡♥♦t❛r❡♠♦s ♠❞❝(F6, F7)♣♦r (F6, F7)✳

Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳✶ ❉♦✐s t❡r♠♦s ❝♦♥s❡❝✉t✐✈♦s ❞❛ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✳

Pr♦✈❛ ▼♦str❛r❡♠♦s✱ ♣♦r ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✱ q✉❡ (Fn+1, Fn) = 1✳ ❉❡ ❢❛t♦✱

♣❛r❛ n = 1✱ t❡♠♦s q✉❡✿

(F2, F1) = (1,1) = 1✳

❙✉♣♦♥❤❛♠♦s q✉❡ ♦ r❡s✉❧t❛❞♦ s❡❥❛ ✈á❧✐❞♦ ♣❛r❛ ❛❧❣✉♠ n✱ ✐st♦ é✱ (Fn+1, Fn) = 1✳

❚❡♠♦s✱ ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✱ q✉❡✿

(Fn+2, Fn+1) = (Fn+2−Fn+1, Fn+1) = (Fn, Fn+1) = 1✱ ♣r♦✈❛♥❞♦✱ ❛ss✐♠✱ ♦ r❡s✉❧t❛❞♦✳

Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳✷ ❆ s♦♠❛ ❞❡ s❡✐s ♥ú♠❡r♦s ❝♦♥s❡❝✉t✐✈♦s ❞❡ ❋✐❜♦♥❛❝❝✐ é ❞✐✈✐sí✈❡❧ ♣♦r ✹✳

❱❡❥❛♠♦s✱ ♣❛r❛ n_≥0 ✭❝♦♠ n ✜①♦✮

5

X

r=0

Fn+r=Fn+Fn+1+Fn+2+Fn+3+Fn+4+Fn+5 = 4Fn+4

Pr♦♣r✐❡❞❛❞❡s ❈❛♣ít✉❧♦ ✷

Pr♦✈❛ P❛r❛ n _≥0✱ t❡♠♦s✿

5

X

r=0

Fn+r = Fn+Fn+1+Fn+2+Fn+3+Fn+4+Fn+5

= (Fn+Fn+1) +Fn+2+Fn+3+Fn+4+ (Fn+3+Fn+4)

= 2Fn+2+ 2Fn+3+ 2Fn+4

= 2(Fn+2+Fn+3) + 2Fn+4

= 4Fn+4

Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳✸ ❆ s♦♠❛ ❞❡ q✉❛✐sq✉❡r ❞❡③ ♥ú♠❡r♦s ❝♦♥s❡❝✉t✐✈♦s ❞❡ ❋✐❜♦♥❛❝❝✐ é ❞✐✈✐sí✈❡❧ ♣♦r ✶✶✳

❉❡ ❢❛t♦✱ ♣❛r❛ n_≥0 ✭❝♦♠ n ✜①♦✮✱ t❡♠♦s✿

9

X

r=0

Fn+r= 11Fn+6✳

Pr♦✈❛ P❛r❛ n _≥0✱ t❡♠♦s✿

9

X

r=0

Fn+r = Fn+Fn+1+Fn+2+Fn+3+Fn+4+Fn+5+Fn+6+Fn+7+Fn+8+Fn+9

= (Fn+Fn+1+Fn+2+Fn+3+Fn+4+Fn+5) +Fn+6+Fn+7+Fn+8+ (Fn+7+Fn+8)

= 4Fn+4+Fn+6+ 2Fn+7+ 2Fn+8

= (4Fn+4+ 4Fn+5) + 7Fn+6

= 4Fn+6+ 7Fn+6

= 11Fn+6

Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳✹ P❛r❛n _≥0✱ n

X

r=0

Fr =Fn+2−1✳

❊st❛ ♣r♦♣r✐❡❞❛❞❡ ❢♦✐ ❞❡s❝♦❜❡rt❛ ♣♦r ❊❞♦✉❛r❞ ▲✉❝❛s ❡♠ ✶✽✼✻✳ ❱❡❥❛♠♦s ❛s s❡❣✉✐♥t❡s s♦♠❛s✿

F0+F1+F2 = 2 = 3−1 =F4−1

F0+F1+F2+F3 = 4 = 5−1 = F5−1

F0+F1+F2+F3+F4 = 7 = 8−1 =F6−1✳ ❊ss❡s r❡s✉❧t❛❞♦s ❛♣♦♥t❛♠ ♣❛r❛ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦✳ ❊♥tã♦ ✈❡❥❛♠♦s✿

Pr♦✈❛ P❛r❛ ♣r♦✈❛r ❡st❛ ♣r♦♣r✐❡❞❛❞❡✱ ✈❛♠♦s ✉t✐❧✐③❛r ❛ ❞❡✜♥✐çã♦ r❡❝✉rs✐✈❛ ❞♦s ♥ú♠❡✲ r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳ ❈♦♥s✐❞❡r❡✿

Pr♦♣r✐❡❞❛❞❡s ❈❛♣ít✉❧♦ ✷

F0 =F2−F1 ❋1 =F3−F2 ❋2 =F4−F3

✳✳✳ ✳✳✳ ✳✳✳ ❋n−1 =Fn+1−Fn

❋n=Fn+2−Fn+1 ❆♦ s♦♠❛r♠♦s ❛s n+ 1 ❡q✉❛çõ❡s✱ ♦❜t❡♠♦s✿

n

X

r=0

Fr = (F2−F1) + (F3−F2) +· · ·+ (Fn+1−Fn) + (Fn+2−Fn+1)

= ₋F1+ (F2−F2) + (F3−F3) +· · ·+ (Fn−Fn) + (Fn+1−Fn+1) +Fn+2

= Fn+2−F1

= Fn+2−1

Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳✺ P❛r❛n _≥0,

n

X

r=0

F_r2 =Fn ·❋n+1

❯t✐❧✐③❛r❡♠♦s ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ ♣❛r❛ ♣r♦✈❛r ❡st❛ ♣r♦♣r✐❡❞❛❞❡✳ ❉❡ ❢❛t♦✱ ♣❛r❛ n = 0✱

t❡♠♦s❀

0

X

r=0

F2

r =F02 = 02 = 0 = 0·✶❂❋0 ·❋1 =F0 ·❋0+1 q✉❡ ❛ ♣r♦♣r✐❡❞❛❞❡ é ✈❡r❞❛❞❡✐r❛✳

❙✉♣♦♥❤❛ q✉❡ s❡❥❛ ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♠ n=k,(k _≥0)✱ ❝♦♠n ✜①♦ ❡ ❛r❜✐trár✐♦✳

❉❛í✱ t❡♠♦s✿

k

X

r=0

F_r2 =FkFk+1✳ ❱♦❧t❛♥❞♦ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ n =k+ 1(≥1)✱ t❡♠♦s✿ k+1

X

r=0

F2

r = k

X

r=0

F2

r

!

+Fk+1

= (FkFk+1) +Fk2+1

= Fk+1(Fk+Fk+1)

= Fk+1Fk+2.

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ r❡s✉❧t❛❞♦ ✈❡r❞❛❞❡✐r♦ ♣❛r❛ n = k+ 1 ❞❡❝♦rr❡ ❞❡ n = k✳ ❉❛í✱

♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✱ ♦ r❡s✉❧t❛❞♦ é ✈❡r❞❛❞❡✐r♦ ♣❛r❛ t♦❞♦ n _≥0✳

Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳✻ ❆ s♦♠❛ ❞♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❞❡ ♦r❞❡♠ í♠♣❛r é ✐❣✉❛❧ ❛ F2n✳

Pr♦♣r✐❡❞❛❞❡s ❈❛♣ít✉❧♦ ✷

P❛r❛ n_≥1,

n

X

r=1

F2r−1 =F1+F3+· · ·+F2n−1 =F2n✳

Pr♦✈❛ P❛r❛ ♣r♦✈❛r ❡st❛ ♣r♦♣r✐❡❞❛❞❡✱ ✈❛♠♦s ✉t✐❧✐③❛r ❛ ❞❡✜♥✐çã♦ ❞❡ s❡q✉ê♥❝✐❛ r❡❝✉r✲ s✐✈❛✿

F1 =F2 ❋3 =F4−F2 ❋5 =F6−F4

✳✳✳ ✳✳✳ ✳✳✳

❋2n−1 =F2n−F2n−2 ❙♦♠❛♥❞♦ ❛s ❡q✉❛çô❡s ♦❜t✐❞❛s✱ t❡r❡♠♦s✿

F1+F3+· · ·+F2n−1 = (F2−F2) + (F4−F4) + (F6 −F6) +· · ·+ (F2n−2−F2n−2) +F2n n

X

r=1

F2r−1 = F2n.

Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳✼ P❛r❛n _≥1,

n

X

r=1

F2r =F2+F4+· · ·+F2r=F2n+1−1✳ Pr♦✈❛ P❛r❛ ♣r♦✈❛r ❡st❛ ♣r♦♣r✐❡❞❛❞❡✱ ✉t✐❧✐③❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ♥❛s Pr♦♣r✐❡❞❛❞❡s ✷✳✷✳✹ ❡ ✷✳✷✳✻✳ ❱❡❥❛♠♦s✿

P❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳✹✱ t❡♠♦s q✉❡ ❛ s♦♠❛ ❞❡ t♦❞♦s ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❛té ❛ ♦r❞❡♠ 2n é✿

F1+F2+F3+· · ·+F2n−1 +F2n=F2n+2−1 ✭✐✮

❡ ♣❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳✻✱ ❛ s♦♠❛ ❞♦s ♥ú♠❡r♦s ❞❡ ♦r❞❡♠ í♠♣❛r ❛té 2n₋1 é✿

F1+F3+F5+· · ·+F2n−1 =F2n ✭✐✐✮

❋❛③❡♥❞♦ ✭✐✮✲✭✐✐✮✱ t❡♠♦s✿

F2+F4+F6+F8+· · ·+F2n=F2n+2−F2n−1 ✭✐✐✐✮

❈♦♠♦ F2n+2 =F2n+1+F2n✱ s✉❜st✐t✉✐♠♦s ❡♠ ✭✐✐✐✮✱ ♦❜t❡♠♦s✿ n

X

r=1

F2r =F2n+1−1✳

Pr♦♣r✐❡❞❛❞❡ ✷✳✷✳✽ P❛r❛n _≥1, Fn−1Fn+1−Fn2 = (−1)n✳

❊st❛ ♣r♦♣r✐❡❞❛❞❡ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❢ór♠✉❧❛ ❞❡ ❈❛ss✐♥✐✳

❋ór♠✉❧❛ ❞❡ ❇✐♥❡t ❈❛♣ít✉❧♦ ✷

Pr♦✈❛ Pr♦✈❛r❡♠♦s ♣♦r ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✳ ❉❡ ❢❛t♦✱ ♣❛r❛ n = 1✱ t❡♠♦s✿

F0F2−F12 = 0.1−12 =−1.

❙✉♣♦♥❤❛♠♦s q✉❡ s❡❥❛ ✈á❧✐❞❛ ♣❛r❛ ❛❧❣✉♠ n _≥1✳ Pr♦✈❡♠♦s q✉❡ t❛♠❜é♠ é ✈á❧✐❞❛ ♣❛r❛

n+ 1✳

FnFn+2−Fn2+1 = Fn(Fn+1+Fn)−Fn2+1

= Fn+1(Fn−Fn+1) +Fn2 = Fn+1(−Fn−1) +Fn2 = ₋(Fn−1Fn+1−Fn2) = ₋(₋1)n = (₋1)n+1

✷✳✸ ❋ór♠✉❧❛ ❞❡ ❇✐♥❡t

❏á ✈✐♠♦s q✉❡ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ❛ s❡q✉ê♥❝✐❛ ❢♦r♠❛❞❛ ♣❡❧♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦✲ ♥❛❝❝✐✱ ✉t✐❧✐③❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛✱ ❞❛❞❛ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧✿

F0 = 0, F1 = 1 ✭❈♦♥❞✐çã♦ ✐♥✐❝✐❛❧✮

Fn =Fn−1+Fn−2, n≥2 ✭❘❡❧❛çã♦ ❞❡ ❘❡❝♦rrê♥❝✐❛✮ ✳

❆❣♦r❛✱ q✉❡r❡♠♦s ❡♥❝♦♥tr❛r ✉♠ t❡r♠♦ q✉❛❧q✉❡r ❞❛ s❡q✉ê♥❝✐❛ s❡♠✱ ♥❡❝❡ss❛r✐❛✲ ♠❡♥t❡✱ ❝❛❧❝✉❧❛r t♦❞♦s ♦s t❡r♠♦s ❛♥t❡r✐♦r❡s✱ ♦✉ s❡❥❛✱ q✉❡r❡♠♦s ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ t❡r♠♦ ❣❡r❛❧ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❡♠ ❢✉♥çã♦ ❞♦ Fn✳ P❛r❛ ✐ss♦✱

✉t✐❧✐③❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ r❡❝♦rrê♥❝✐❛✳

P❛r❛ ❝♦♥st❛♥t❡s r❡❛✐s C0, C1, C2,· · · , Ck, ❝♦♠ C0 6= 0 ❡Ck 6= 0✱ ✉♠❛ ❡①♣r❡ssã♦

❞❛ ❢♦r♠❛✿

C0an+C1an−1+C2an−2+· · ·+Ckan−k= 0✱ ♦♥❞❡n✱

é ❝❤❛♠❛❞♦ ♦ k✲és✐♠♦ t❡r♠♦ ❞❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ ❤♦♠♦❣ê♥❡❛ ❧✐♥❡❛r ❝♦♠ ❝♦❡✜✲

❝✐❡♥t❡s ❝♦♥st❛♥t❡s✳

❈♦♥s✐❞❡r❡ ♦ ❝❛s♦ ❡♠ q✉❡ k = 2, C0 = 1 ❡ C2 6= 0✳ P♦r ❡①❡♠♣❧♦✿

an−5an−1+ 6an−2 = 0,♦✉ ❛✐♥❞❛✱ ❛n = 5an−1−6an−2✱

♦❜t❡♠♦s ❛í✱ ✉♠❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ ❧✐♥❡❛r ❤♦♠♦❣ê♥❡❛✱ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥✲ t❡s✳

❋ór♠✉❧❛ ❞❡ ❇✐♥❡t ❈❛♣ít✉❧♦ ✷

❙❡❥❛ an=Arn✱ ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛✱ ♦♥❞❡ A❡ r sã♦ ❝♦♥st❛♥t❡s ❞✐❢❡r❡♥t❡s ❞❡

③❡r♦✳ ❙✉❜st✐t✉✐♥❞♦ ♥❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ ❡♥❝♦♥tr❛❞❛✱ t❡♠♦s✿

Arn_{= 5Ar}n−1_−6Arn−2 ❉✐✈✐❞✐♥❞♦ ♣♦r Arn−2_{✱ t❡♠♦s✿}

r2 = 5r₋6_↔r2₋5r+ 6 = 0 ✭✷✳✶✮

✉♠❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❡♠ r✳ ❊st❛ ❡q✉❛çã♦ é ❝❤❛♠❛❞❛ ❞❡ ❡q✉❛çã♦ ❝❛r❛❝✲

t❡ríst✐❝❛✳ P❛r❛ ❛ ❡q✉❛çã♦ r2 _{−5r + 6 = 0✱ ❛s r❛í③❡s sã♦ r = 2 ❡ r = 3 ✭r❛í③❡s} ❝❛r❛❝t❡ríst✐❝❛s✮✳

❯♠❛ s♦❧✉çã♦ ❣❡r❛❧ ♣❛r❛

an = 5an−1−6an−2 t❡♠ ❛ ❢♦r♠❛

an=c12n+c23n✱

♦♥❞❡ c1 ❡ c2 sã♦ ❝♦♥st❛♥t❡s ❛r❜✐trár✐❛s✳ ❙✉❜st✐t✉✐♥❞♦ ❛ s♦❧✉çã♦ ❣❡r❛❧ ♥❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ an−5an−1+ 6an−2 = 0✱ ♥ós ❡♥❝♦♥tr❛♠♦s✿

(c12n+c23n)−5(c12n−1+c23n−1) + 6(c12n−2+c23n−2) = 0

=c12n−2(22−5(2) + 6) +c23n−2(32−5(3) + 6)

=c12n−2(0) +c23n−2(0) = 0

❱❡r✐✜❝❛♠♦s ❞✐r❡t❛♠❡♥t❡ q✉❡ an = c12n+c23n é ❞❡ ❢❛t♦ ✉♠❛ s♦❧✉çã♦ ❣❡r❛❧✳ ▼♦s✲ tr❛r❡♠♦s ♦ ✈❛❧♦r ❞❡ an ♣❛r❛ ❞♦✐s ✈❛❧♦r❡s ❡s♣❡❝í✜❝♦s ❞❡ a0 ❡ a1✱ ❝♦♠ ❡ss❡s ✈❛❧♦r❡s ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r c1 ❡c2✳

❚♦♠❡♠♦s ❝♦♠♦ ❡①❡♠♣❧♦ a0 = 0 ❡ a1 = 4✳

1 = a0 =c120+c230 =c1+c2 ✹❂❛1 =c121+c231 = 2c1+ 3c2✱

❘❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛ ❡♥❝♦♥tr❛❞♦✱ ♦❜t❡♠♦s c1 =−1 ❡c2 = 2✳ ❊♥tã♦✱

an= (−1)2n+ 2.3n, n≥0✱

é s♦❧✉çã♦ ✭ú♥✐❝❛✮ ❞❛ ❡q✉❛çã♦ ✐♥✐❝✐❛❧✿

❋ór♠✉❧❛ ❞❡ ❇✐♥❡t ❈❛♣ít✉❧♦ ✷

❚❛♠❜é♠ é s♦❧✉çã♦ ✭ú♥✐❝❛✮ ❞❛ ❡q✉❛çã♦ ❡q✉✐✈❛❧❡♥t❡✿

an+2−5an+1+ 6an = 0, n≥2, a0 = 1, a1 = 4✳

P❛r❛ r❡s♦❧✈❡r ❛ r❡❝♦rrê♥❝✐❛✱ é ♥❡❝❡ssár✐♦ q✉❡ ❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ♦❜t✐❞❛ ❡♠

r ♣♦ss✉❛ ❞✉❛s r❛í③❡s r❡❛✐s ❞✐st✐♥t❛s✳

❈♦♥s✐❞❡r❡ ❛❣♦r❛ ❛ r❡❝♦rrê♥❝✐❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✿

Fn =Fn−1+Fn−2, n≥2, F0 = 0, F1 = 1✳

❯t✐❧✐③❛♥❞♦ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ♣❛r❛ ♦ ♣r✐♠❡✐r♦ ❝❛s♦ q✉❡ ♠♦str❛♠♦s✱ ✈❛♠♦s s✉❜st✐t✉✐r

Fn=Arn, A6= 0, r6= 0✳ ❙✉❜st✐t✉✐♥❞♦ ♥❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ ❡♥❝♦♥tr❛♠♦s✿

Arn_=Arn−1_+Arn−2

❉✐✈✐❞✐♥❞♦ ♣♦r Arn−2_{✱ ❡♥❝♦♥tr❛♠♦s ❛ ❡q✉❛çã♦ ❝❛r❛❝t❡ríst✐❝❛}

r2_{−r−1 = 0✳}

❘❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ♦❜t❡♠♦s ❛s r❛í③❡s ❝❛r❛❝t❡ríst✐❝❛s✿

r = −(−1)±

√

(−1)2_{−4(1)(−1)}

2(1) =

1±√5 2

❆s r❛í③❡s ❝❛r❛❝t❡ríst✐❝❛s ♦❜t✐❞❛s ❞❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ sã♦✿

α= 1+√5

2 ❡ β = 1−√5

2 ✳

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

Fn=c1αn+c2βn, n≥0✱ ❝♦♠

0 = F0 =c1+c2 ❡

1 =F1 =c1α+c2β =c1

1+√5 2

+c2

1−√5 2

✱

❙❡❣✉❡✲s❡ q✉❡ c1 = √1₅ ❡ c2 = −√1₅✳ ❉❛í✱ ♣♦❞❡♠♦s ❡①♣r❡ss❛r Fn ♣♦r✿

Fn= √1₅αn− √1₅βn, n≥0✳

❊♥tr❡ ❛s ♠✉✐t❛s ♣r♦♣r✐❡❞❛❞❡s s❛t✐s❢❡✐t❛s ♣♦r α ❡β✱ t❡♠♦s✿

❋ór♠✉❧❛ ❞❡ ❇✐♥❡t ❈❛♣ít✉❧♦ ✷

α2 _{=α+ 1}

αβ =₋1

α₋β =√5

α2_+β2 _{= 3}

α−1 _=−β

β2 _{=β+ 1}

α+β = 1

β−1 _=−α

α2_−β2 ₌√₅

❈♦♠♦ α₋β =√5✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❢ór♠✉❧❛ ♣❛r❛ Fn ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

Fn= α

n

−βn

α−β , n ≥0✳

❊st❛ r❡♣r❡s❡♥t❛çã♦ ♣❛r❛ Fn é ❝❤❛♠❛❞❛ ❞❡ ❋ór♠✉❧❛ ❞❡ ❇✐♥❡t ♣❛r❛ ♦s ♥ú♠❡r♦s ❞❡

❋✐❜♦♥❛❝❝✐✳

❯t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ❇✐♥❡t✱ ✈❡❥❛♠♦s ❞✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡♥✈♦❧✈❡♥❞♦ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳

Pr♦♣r✐❡❞❛❞❡ ✷✳✸✳✶

lim n→∞

Fn+1

Fn =α.

Pr♦✈❛ ❙❡♥❞♦ α= (1 +√5)/2 ❡β = (1₋√5)/2✱ s❡❣✉❡✲s❡ q✉❡

|β/α_| = (1−

√

5)/(1 +√5)

< 1✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❝♦♠♦ n → ∞✱ t❡♠♦s

|β/α_|n _{→0❡ (β/α)}n _{→0✳ P♦rt❛♥t♦✿} lim

n→∞

Fn+ 1

Fn

= lim n→∞

(αn+1_−βn+1_)/(α−β)

(αn_−βn_)/(α−β) = lim_n→∞

αn+1_−βn+1

αn_−βn = lim

n→∞

α₋β(β/α)n 1₋(β/α)n =

α₋β(0) 1₋0 =α.

Pr♦♣r✐❡❞❛❞❡ ✷✳✸✳✷

n

X

k=0

n k

2kFk =F3n

❖❜s❡r✈❡ q✉❡ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ❡♥✈♦❧✈❡ ♥ú♠❡r♦s ❜✐♥♦♠✐❛✐s✱ ♣♦r ✐ss♦✱ ❛♥t❡s ❞❡ ♣r♦✈❛r ❡st❛ ♣r♦♣r✐❡❞❛❞❡✱ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦✿ P❛r❛ ❛s ✈❛r✐á✈❡✐s x, y r❡❛✐s ❡ n✱ ✉♠

❋✐❜♦♥❛❝❝✐ ❡ ❛s ■❞❡♥t✐❞❛❞❡s ❞❡ ▲✉❝❛s ❈❛♣ít✉❧♦ ✷

✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦✱ t❡♠♦s✿

(x+y)n =

n

0

x0yn+

n

1

x1yn−1₊

· · ·+

n n₋1

xn−1_y1₊

n n

xny0

= n X k=0 n k

xkyn−k

❖♥❞❡✱

n k

= n!

k!(n₋k)!, 0! = 1

Pr♦✈❛ P❛r❛ ♣r♦✈❛r ❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♥s✐❞❡r❡ ❛ ❢ór♠✉❧❛ ❞❡ ❇✐♥❡t ❡ ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s✿

α2 _{=α+ 1 ❡}

2α+ 1 = (α+ 1) +α=α2_{+α=α(α+ 1) =α(α}2_{) = α}3_✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ 2β+ 1 =β3_{✳ ❱♦❧t❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡✱ t❡♠♦s✿}

n X k=0 n k

2kFk = n X k=0 n k 2k

αk_−βn

α₋β

= 1

α₋β

n X k=0 n k

2kαk₋ 1 α₋β

n X k=0 n k

2kβk

= 1

α₋β

n X k=0 n k

(2α)k₁n−k₋ 1

α₋β

n X k=0 n k

(2β)k₁n−k

= 1

α₋β(2α+ 1

n

− _α 1

−β(2β+ 1)

n

= 1

α₋β(α

3₎n

− _α 1 −β(β

3₎n

= α

3n

β3n =F3n.

✷✳✹ ❋✐❜♦♥❛❝❝✐ ❡ ❛ ❙❡q✉ê♥❝✐❛ ❞❡ ▲✉❝❛s

❯t✐❧✐③❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ ♣❛r❛ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ ❞✐❢❡r❡♥t❡s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ♥♦✈❛s s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛s✳ ❈♦♥s✐❞❡r❡ Ln ♦

❋✐❜♦♥❛❝❝✐ ❡ ❛s ■❞❡♥t✐❞❛❞❡s ❞❡ ▲✉❝❛s ❈❛♣ít✉❧♦ ✷

L3 = 1 + 3 = 4 ▲4 = 3 + 4 = 7

▲5 = 4 + 7 = 11

❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s 1,3,4,7,11, ... ❝♦rr❡s♣♦♥❞❡♠ ❛♦s t❡r♠♦s ❞❛ ❙❡q✉ê♥❝✐❛ ❞❡

▲✉❝❛s✳

❖s ♥ú♠❡r♦s ❞❡ ▲✉❝❛s ♣♦❞❡♠ s❡r ❞❛❞♦s ♣♦r✿

Ln =Fn+1+Fn−1, n≥1✳

P♦❞❡♠♦s ❞✐③❡r q✉❡ ❡ss❛ é ❛ ♣r✐♠❡✐r❛ ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ❞❡ ▲✉❝❛s✳ ❖❜s❡r✈❡ q✉❡ ❤á ✉♠❛ r❡❧❛çã♦ ❞♦s ♥ú♠❡r♦s ❞❡ ▲✉❝❛s ❝♦♠ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳

❱❡❥❛♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ r❡❧❛❝✐♦♥❛♠ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ ♦s ♥ú♠❡r♦s ❞❡ ▲✉❝❛s✳

Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✶ ❉❡✜♥✐çã♦ r❡❝✉rs✐✈❛ ❞♦s ◆ú♠❡r♦s ❞❡ ▲✉❝❛s

Ln=Fn+1+Fn−1, n ≥1✳

Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✷ Ln =Fn+2−Fn−2, n ≥2✳

❯t✐❧✐③❛♥❞♦ ❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✶ t❡♠♦s q✉❡ Ln =Fn+1+Fn−1✱ ❡ ♣❡❧❛ s❡q✉ê♥❝✐❛

❞❡ ❋✐❜♦♥❛❝❝✐ s❛❜❡♠♦s q✉❡ Fn+1 =Fn+2−Fn ❡Fn−1 =Fn−Fn−2✱ s✉❜st✐t✉✐♥❞♦ ♥❛

Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✶✱ t❡♠♦s✿

Ln = Fn+1+Fn−1

= (Fn+2−Fn) + (Fn−Fn−2)

= Fn+2−Fn−2, n≥2. Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✸ Fn+Ln= 2Fn+1 ♣❛r❛ n≥0✳

P❛r❛ ♣r♦✈❛r ❡st❛ ♣r♦♣r✐❡❞❛❞❡ ✉t✐❧✐③❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ r❡❝✉rs✐✈❛ ❞♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ ❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✶✳ ❱❡❥❛♠♦s✿

Fn+Ln = (Fn+1−Fn−1) + (Fn+1+Fn−1) = 2Fn+1 Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✹ 2Ln+1−Ln= 5Fn, n ≥0✳

❯t✐❧✐③❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ✷✳✹✳✸✱Ln = 2Fn+1−Fn✱ ❡ Ln+1 = 2Fn+2−Fn+1✱ ❞❛í✱

♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r✿

❋✐❜♦♥❛❝❝✐ ❡ ❛s ■❞❡♥t✐❞❛❞❡s ❞❡ ▲✉❝❛s ❈❛♣ít✉❧♦ ✷

2Ln+1−Ln = 2(2Fn+2−Fn+1)−(2Fn+1−Fn) = 4Fn+2−2Fn+1−2Fn+1+Fn = 4(Fn+2−Fn+1) +Fn

= 4Fn+Fn= 5Fn

Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✺

L2

n−5Fn2 = 4(−1)n, ♣❛r❛ n≥0

Pr♦✈❛r❡♠♦s ❡st❛ ♣r♦♣r✐❡❞❛❞❡ ✉t✐❧✐③❛♥❞♦ ❛s Pr♦♣r✐❡❞❛❞❡s ✷✳✹✳✶ ❡ ✷✳✹✳✸ ❡ ❛ ❋ór♠✉❧❛ ❞❡ ❈❛ss✐♥✐✳

❊♥tã♦ ✈❡❥❛♠♦s✿

Fn−1Fn+1−Fn2 = (−1)n (Ln−Fn+1)

Fn+Ln 2

−F_n2 = (₋1)n

Ln−

Fn+Ln 2

Fn+Ln 2

−F_n2 = (₋1)n

Ln−Fn 2

Fn+Ln 2

−F_n2 = (₋1)n

L2_n−F_n2₋4F_n2 = 4(₋1)n

L2n−5Fn2 = 4(−1)n

❚❡♦r❡♠❛ ✷✳✹✳✶ P❛r❛ n_≥0✱

Fn+1 =

Fn+

p

5F2

n+ 4(−1)n

2 ❡ Ln+1 =

Ln+

p

5[L2

n+ 4(−1)n] 2

Pr♦✈❛ ❉❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✸ ❥á s❛❜❡♠♦s q✉❡

Fn+Ln = 2Fn+1✱ ❧♦❣♦ 2Fn+1 −Fn = Ln✳ ❙✉❜st✐t✉✐♥❞♦ ♥❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✺

♦❜t❡♠♦s✿

❋✐❜♦♥❛❝❝✐ ❡ ❛s ■❞❡♥t✐❞❛❞❡s ❞❡ ▲✉❝❛s ❈❛♣ít✉❧♦ ✷

(2Fn+1−Fn)2−5Fn2 = 4(−1)n

(2Fn+1−Fn)2 = 5Fn2+ 4(−1)n 2Fn+1−Fn = ±

p

5F2

n + 4(−1)n

Fn+1 =

Fn±

p

5F2

n+ 4(−1)n 2

❈♦♠♦ Fn ≥0✱ t❡r❡♠♦s✿

Fn+1 =

Fn+

p

5F2

n + 4(−1)n 2

Pr♦✈❛♠♦s ❛ss✐♠ ❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞♦ t❡♦r❡♠❛✳ ❱❡❥❛♠♦s ❛ s❡❣✉♥❞❛ ♣❛rt❡✿

❉❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✹ s❛❜❡♠♦s q✉❡✿ 2Ln+1−Ln= 5Fn✱ ❡♥tã♦ Fn = 2Ln+1−L

n

5 ❙✉❜st✐t✉✐♥❞♦ ♥❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✺✱ ♦❜t❡♠♦s✿

L2_n−5

2Ln+1−Ln 5

2

= 4(₋1)n

L2_n− (2Ln+1−Ln)

2

5 = 4(−1)

n

(2Ln+1−Ln)2 = 5[L2n+ 4(−1)n] 2Ln+1 = Ln+

p

5[L2

n+ 4(−1)n]

Ln+1 =

Ln+

p

5[L2

n+ 4(−1)n] 2

❚❡♦r❡♠❛ ✷✳✹✳✷ ●❡♥❡r❛❧✐③❛çã♦ ❞❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❈❛ss✐♥✐ P❛r❛n _≥r >0✱

Fn+rFn−r−Fn2 = (−1)n+r+1Fr2

❆♥t❡s ❞❡ ❣❡♥❡r❛❧✐③❛r ❛ ❢ór♠✉❧❛ ❞❡ ❈❛ss✐♥✐✱ ✈❛♠♦s ✉t✐❧✐③❛r ❛ ❢ór♠✉❧❛ ❞❡ ❇✐♥❡t ♣❛r❛ ♦s ♥ú♠❡r♦s ❞❡ ▲✉❝❛s✳

P❛r❛n _≥0✱ t❡♠♦s✿

L2_n = (αn+βn)2 =α2n+β2n+ 2(αβ)n = L2n+ 2(−1)n, ❞❡s❞❡ q✉❡ αβ =−1

❋❡r♠❛t ❡ ❋✐❜♦♥❛❝❝✐ ❈❛♣ít✉❧♦ ✷

❉❛ Pr♦♣r✐❡❞❛❞❡ ✷✳✹✳✺✱ t❡♠♦s

5F2

n =L2n−4(−1)n = [L2n+ 2(−1)n]−4(−1)n =L2n−2(−1)n

♦✉ 5F2

n+ 2(−1)n =L2n

Pr♦✈❛ ❯s❛♥❞♦ ❛ ❋ór♠✉❧❛ ❞❡ ❇✐♥❡t ♣❛r❛ ♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ s✉❜st✐t✉✐♥❞♦ ♥♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ ❞❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❈❛ss✐♥✐✿

Fn+rFn−r−Fn2 =

= α

n+r_−βn+r

α₋β

αn−r_βn−r

α₋β −

αn_−βn

α₋β

2

= α

2n_−αn+r_βn−r_−αn−r_βn+r_+β2n

(α₋β)2 −

α2n_−2(αβ)n_+β2n (α₋β)2

❈♦♠♦ (α₋ β)2 _{= (}√₅₎2 _{= 5✱ ❡ (αβ) = −1✱ t❡♠♦s q✉❡ α}−_{1 = −β, β}−_{1 = −α✳}

❋❛③❡♥❞♦ ✉s♦ ❞❡ss❛s r❡❧❛çõ❡s✱ t❡♠♦s

= −(αβ)

n_αr_β−r_−(αβ)n_α−r_βr_{+ 2(−1)}n 5

= −(−1)

n_αr_(β−1₎r₋₍₋₁₎n_(α−1₎r_βr_{+ 2(−1)}n 5

= −(−1)

n_αr_(−α)r₋₍₋₁₎n_(−β)r_βr_{+ 2(−1)}n 5

= −(−1)

n₍₋₁₎r_α2r₋₍₋₁₎n₍₋₁₎r_β2r_{+ 2(−1)}n 5

= (−1)

n+r+1_[α2r_+β2r_{] + 2(−1)}n 5

= (−1)

n+r+1_L

2r+ 2(−1)n 5

= (−1)

n+r+1_[5F2

r + 2(−1)r] + 2(−1)n 5

= (₋1)n+r+1F_r2+

2 5

[(₋1)n+2r+1+ (₋1)n]

= (₋1)n+r+1F_r2+

2 5

[(₋1)n+1+ (₋1)n]

= (₋1)n+r+1F_r2

❋❡r♠❛t ❡ ❋✐❜♦♥❛❝❝✐ ❈❛♣ít✉❧♦ ✷

✷✳✺ ❋❡r♠❛t ❡ ❋✐❜♦♥❛❝❝✐

❖ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês P✐❡rr❡ ❞❡ ❋❡r♠❛t ♦❜s❡r✈♦✉ q✉❡ ♦s ♥ú♠❡r♦s 1,3,8,120 tê♠

✉♠❛ ✐♥t❡r❡ss❛♥t❡ ♣r♦♣r✐❡❞❛❞❡✳

❯♠ ♠❛✐s ♦ ♣r♦❞✉t♦ ❞❡ q✉❛✐sq✉❡r ❞♦✐s ❞❡❧❡s é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦

1 + 1.3 = 22 ✶✰ ✶✳✽❂✸2 ✶✰✶✳✶✷✵❂✶✶2

✶✰✸✳✽❂✺2 ✶✰✸✳✶✷✵❂✶✾2 ✶✰✽✳✶✷✵❂✸✶2

❊♠ ✶✾✻✾✱ ❆❧❛♥ ❇❛❦❡r ❡ ❍❛r♦❧❞ ❉❛✈❡♥♣♦rt ❞♦ ❚r✐♥✐t② ❈♦❧❧❡❣❡✱ ❡♠ ❈❛♠❜r✐❞❣❡ ♣r♦✈❛r❛♠ q✉❡ s❡ ♦s ♥ú♠❡r♦s 1,3,8, ❡ x tê♠ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡✱ ❡♥tã♦ x ❞❡✈❡ s❡r

120✳ ❈✉r✐♦s❛♠❡♥t❡✱ ♦❜s❡r✈❡ q✉❡ 1 = F2,3 =F4,8 =F6 ❡ 120 = 4.2.3.5 = 4F3F4F5✳ ❆ss✐♠✱ ♦✐t♦ ❛♥♦s ♠❛✐s t❛r❞❡✱ ❱✳ ❍♦❣❣❛tt✱ ❏r✳✱ ❡ ●❊ ❇❡r❣✉♠ ❞♦ ❙✉❧ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ ❞❡ ❉❛❦♦t❛✱ ❛ ♣❛rt✐r ❞❡ss❛ ♦❜s❡r✈❛çã♦✱ ❡st❛❜❡❧❡❝❡r❛♠ ❛ s❡❣✉✐♥t❡ ❣❡♥❡r❛❧✐③❛✲ çã♦✿

❚❡♦r❡♠❛ ✷✳✺✳✶ ❖s ♥ú♠❡r♦sF2n, F2n+2, F2n+4 ❡4F2n+1F2n+2F2n+3 tê♠ ❛ ♣r♦♣r✐✲ ❡❞❛❞❡ ❞❡ ✉♠ ♠❛✐s ♦ ♣r♦❞✉t♦ ❞❡ q✉❛✐sq✉❡r ❞♦✐s ❞❡❧❡s é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✳

Pr♦✈❛ P❡❧❛ ❢ór♠✉❧❛ ❞❡ ❈❛ss✐♥✐ t❡♠♦s q✉❡ 1 +F2nF2n+2 = F22n+1✳ ❉❛ ♠❡s♠❛

❢♦r♠❛✱ 1 +F2n+1F2n+3 =F22n+2 ❡ 1 +F2n+2F2n+4 =F22n+3✳ ❊♠ s❡❣✉✐❞❛✱ t❡♠♦s✿

1 +F2n(4F2n+1F2n+2F2n+3)

= 1 + 4(F2nF2n+2)(F2n+1F2n+3)

= 1 + 4(F₂2_n+1−1)(F₂2_n+2+ 1)♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ❈❛ss✐♥✐ = 4F₂2_n+1F₂2_n+2−4(F₂2_n+2−F₂2_n+1)₋3

= 4F₂2_n+1F₂2_n+2−4(F2n+3F2n)−3

= 4F₂2_n+1F₂2_n+2−4F2n+3(F2n+2−F2n+1)−3

= 4F₂2_n+1F₂2_n+2−4F2n+3F2n+2+ 4F2n+1F2n+3−3

= 4F2

2n+1F22n+2−4F2n+3F2n+2+ 4(F22n+2+ 1)−3

= 4F₂2_n+1F₂2_n+2−4F2n+2(F2n+3−F2n+2) + 1

= 4F₂2_n+1F₂2_n+2−4F2n+1F2n+2+1

= (2F2n+1F2n+2−1)2

❈❛♣ít✉❧♦ ✸

❘❡❧❛çã♦ ❞♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐

❝♦♠ ♦✉tr❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛

❊st✉❞❛r❡♠♦s ❛❣♦r❛ ❛ r❡❧❛çã♦ ❞♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❝♦♠ ♦✉tr❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛✱ ❡♥tr❡ ❡❧❛s✱ ❞❡st❛❝❛♠♦s ❛ ❚r✐❣♦♥♦♠❡tr✐❛✱ ❛s ▼❛tr✐③❡s ❡ ❛ ●❡♦♠❡tr✐❛✳ ❆❜♦r❞❛r❡♠♦s ❛ ❊❧✐♣s❡ ❡ ❛ ❍✐♣ér❜♦❧❡ ❞❡ ♦✉r♦✳ ❋❛r❡♠♦s t❛♠❜é♠ ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ❞♦s ♥ú♠❡r♦s ❚r✐❜♦♥❛❝❝✐ ❡ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s

✸✳✶ ❋✐❜♦♥❛❝❝✐ ❡ ❛ ❚r✐❣♦♥♦♠❡tr✐❛

❆ ♣❛rt✐r ❞❡ ❛❧❣✉♠❛s ✐❞❡♥t✐❞❛❞❡s tr✐❣♦♥♦♠étr✐❝❛s ❡ ❛ ❢ór♠✉❧❛ ❞❡ ❇✐♥❡t✱ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r ❛ ❢ór♠✉❧❛ tr✐❣♦♥♦♠étr✐❝❛ ♣❛r❛ ♦s ♥ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳

❊ss❛ ❢ór♠✉❧❛ ❢♦✐ ❡st❛❜❡❧❡❝✐❞❛ ♣♦r ❲✳ ❍♦♣❡✲❏♦♥❡s ❡♠ ✶✾✷✶✳ ❈♦♥s✐❞❡r❡ ❛s ✐❞❡♥t✐❞❛❞❡s tr✐❣♦♥♦♠étr✐❝❛s✿

✶✳ sin 2θ= 2 sinθcosθ

✷✳ cos 2θ = cos2_θ−sin2_{θ= 2 cos}2_θ−1

✸✳ cos 3θ = cos(2θ+θ) = cos 2θcosθ₋sin 2θsinθ

❈♦♥s✐❞❡r❛♥❞♦ ❛s ✐❞❡♥t✐❞❛❞❡s ❡ ✭✶✮ ❡ ✭✷✮ ❡♠ ✭✸✮✱ ♦❜t❡♠♦s✿

cos 3θ = cosθ((2 cos2θ₋1)₋(2 sinθcosθ) sinθ

= 2 cos3θ₋cosθ₋2 cosθsin2θ

= 2 cos3θ₋cosθ₋2 cosθ(1₋cos2θ) = 2 cos3θ₋cosθ+ 2 cos3θ₋2 cosθ

= 4 cos3θ₋3 cosθ

❋✐❜♦♥❛❝❝✐ ❡ ❛ ❚r✐❣♦♥♦♠❡tr✐❛ ❈❛♣ít✉❧♦ ✸

❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ θ =π/10✳ ❚❡♠♦s❀ π

2 = 5θ = 2θ+ 3θ✳

❉❛í✱ t❡♠♦s q✉❡ 2θ ❡ 3θ sã♦ â♥❣✉❧♦s ❝♦♠♣❧❡♠❡♥t❛r❡s ❡✱ sin 2θ =cos3θ✳

2 sinθcosθ = sin 2θ = cos 3θ = 4 cos3_{θ−3 cosθ} ❉✐✈✐❞✐♥❞♦ ♣♦r cosθ ✭❝♦♠ cosθ₆= 0✮✱♦❜t❡♠♦s❀

2 sinθ = 4 cos2θ₋3 2 sinθ = 4(1₋sin2_θ)−3

2 sinθ = ₋4 sin2θ+ 1 ❡,

4 sin2θ+ 2 sinθ₋1 = 0,

✉♠❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❡♠ sinθ✳

❘❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ♦❜t✐❞❛✱ ❡♥❝♦♥tr❛♠♦s ❛s r❛í③❡s✿

sinθ = −2±

√

22_{−4(4)(−1)}

2(4) = − 2±√20

8 = − 1±√5

4

❈♦♠♦θ=π/10✉♠ ❛r❝♦ ❞♦ ♣r✐♠❡✐r♦ q✉❛❞r❛♥t❡✱ t❡♠♦s q✉❡ ♦ sinθ >0✱ t❡♠♦s q✉❡✿ sin₁₀π = sinθ= (−1+

√

5) 4 =−

1 2

₁₋√₅

2

❂✲ 1

2)β = − 1 2

−α1

= ₂1_α✳

❉❡s❞❡ q✉❡ αβ =₋1✳

❈❛❧❝✉❧❛r❡♠♦s ❛❣♦r❛ ♦ cos 2θ✳

❈♦♠♦θ = (π/10),(π/5) = 2θ ❡

cosπ

5 = cos 2θ

= cos2θ₋sin2θ

= 1₋2 sin2θ

= 1₋2

1 2α

2

= 1₋ 1

2α2

= 2α

2₋₁

❋✐❜♦♥❛❝❝✐ ❡ ❛ ❚r✐❣♦♥♦♠❡tr✐❛ ❈❛♣ít✉❧♦ ✸

▼✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s t❡r♠♦s ♣♦r β2_{✱ ♦❜t❡♠♦s}

= (2α

2_)β2

2α2_β2

= 2α

2_β2_−β2

2α2_β2 (♣❛r❛ αβ =−1, β

2 _{=β+ 1)}

= 1−β

2 =

α

2.(❞❡s❞❡ q✉❡ α+β = 1)

❉❛í✱ ❡♥❝♦♥tr❛♠♦s cos(π/5) =α/2)✳ ❈❛❧❝✉❧❛♥❞♦ ♦ cos3π

5 ✱ t❡♠♦s✱

cos3π

5 = cos 3

π

5

= 4 cos3 π

5 −3 cos

π

5

= 4α 2

3

−3α 2

= 1

2α

3

− 3₂α

= 1

2α(α

2

−3)

❈♦♠♦ α2_+β2 _{= 3→α}2_{−3 =−β}2_{✱ t❡♠♦s q✉❡✱}

cos3π

5 =

1

2α(−β)

❈♦♠♦ αβ =₋1, t❡♠♦s q✉❡: cos3π

5 =

1 2

−1

β

(₋β2) cos3π

5 =

1 2β

❏á ❡♥❝♦♥tr❛♠♦s q✉❡cos(π/5) = α/2✱ ❡♥tã♦α = 2 cos(π/5)❡cos(3π/5) = (1/2)β✳

▲♦❣♦ β = 2 cos(3π/5)✳

❙✉❜st✐t✉✐♥❞♦ ❛s r❡❧❛çõ❡s ❡♥❝♦♥tr❛❞❛s ♥❛ ❋ór♠✉❧❛ ❞❡ ❇✐♥❡t✱ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ❛ ❢ór♠✉❧❛ tr✐❣♦♥♦♠étr✐❝❛ ♣❛r❛ ♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✳ ❱❡❥❛♠♦s✿

❋✐❜♦♥❛❝❝✐ ❡ ❛s ▼❛tr✐③❡s ❈❛♣ít✉❧♦ ✸

Fn =

αn_−βn

α₋β =

αn_−βn √

5

= (2 cos(π/5))

n_{−(2 cos(3π/5))}n √

5

= _√1 5(2

n₎

cosnπ 5

−cosn

3π

5

, n_≥0.

✸✳✷ ❋✐❜♦♥❛❝❝✐ ❡ ❛s ▼❛tr✐③❡s

❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ♦♣❡r❛çõ❡s ❝♦♠ ♠❛tr✐③❡s✳ ❆ ♣❛rt✐r ❞❡❧❛s✱ ✈❛♠♦s r❡❧❛❝✐♦♥❛r ♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐ ❛ ✉♠❛ ♠❛tr✐③ ❡s♣❡❝✐❛❧✱ q✉❡ ❢♦✐ ❡st✉❞❛❞❛ ♣♦r ❈❤❛r❧❡s ❍✳ ❑✐♥❣✱ ❡♠ s✉❛ ❚❡s❡ ❞❡ ▼❡str❛❞♦ ❡♠ ✶✾✻✵✱ ❈❛❧✐❢ór♥✐❛✳ ❊❧❡ ❛ ❝❤❛♠♦✉ ❞❡ ▼❛tr✐③ ◗✳

Q=

1 1 1 0

❖❜s❡r✈❡ ♦s r❡s✉❧t❛❞♦s ❛ s❡❣✉✐r✿

Q2 ₌

1 1 1 0 1 1 1 0 = 2 1 1 1

Q3 _=Q.Q2 ₌

1 1 1 0 2 1 1 1 = 3 2 2 1

Q4 _=Q.Q3 ₌

1 1 1 0 3 2 2 1 = 5 3 3 2

Q5 _=Q.Q4 ₌

1 1 1 0 5 3 3 1 = 8 5 5 3

❆ ♣❛rt✐r ❞❛s ♠❛tr✐③❡s ♦❜t✐❞❛s✱ ✈❛♠♦s ❡s❝r❡✈❡r ❝❛❞❛ ✉♠❛ ❞❡❧❛s s✉❜st✐t✉✐♥❞♦ ♦s t❡r♠♦s ♣❡❧♦s ◆ú♠❡r♦s ❞❡ ❋✐❜♦♥❛❝❝✐✿

◗❂ F2 F1

F1 F0

Q2 ₌

F3 F2

F2 F1

Q3 ₌

F4 F3

F3 F2

Q4 ₌

F5 F4

F4 F3

Q5 ₌

F6 F5

F5 F4

P❛rt✐♥❞♦ ❞❡ss❡ r❡s✉❧t❛❞♦✱ ❢♦✐ ❡st❛❜❡❧❡❝✐❞♦ ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿ ❚❡♦r❡♠❛ ✸✳✷✳✶ P❛r❛ ❛ ♠❛tr✐③

Q= 1 1 1 0 ❡

n _≥1✱ t❡♠♦s q✉❡✿

Qn=

Fn+1 Fn

Fn Fn−1

✳