❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
▼❛t❡♠át✐❝❛ ❞✐s❝r❡t❛✿
❚ó♣✐❝♦s ❞❡ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s
❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s✳
♣♦r
❋❆❇■❆◆❖ ❏❖❙➱ ❉❊ ❈❆❙❚❘❖
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❇♦❝❦❡r ◆❡t♦
❉✐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✐❝❛✳
C355m Castro, Fabiano José de.
Matemática discreta: tópicos de recorrências lineares e suas aplicações / Fabiano José de Castro.- João Pessoa, 2016.
77f.
Orientador: Carlos Bocker Neto Dissertação (Mestrado) - UFPB/CCEN
1. Matemática. 2. Matemática discreta. 3. Ensino básico. 4. Sequências. 5. Recorrências. 6. Aplicações.
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛♦ ❉❡✉s ●r❛♥❞✐♦s♦ ♣♦r ♠❡ ❢❛③❡r ❝❛♣❛③ ❛tr❛✈és ❞♦ ❡s✲ ❢♦rç♦ ❡ ❞❡❞✐❝❛çã♦ ❝♦♥❝❧✉✐r ❡st❡ tr❛❜❛❧❤♦✳ ●♦st❛r✐❛ ❞❡ ❤♦♠❡♥❛❣❡❛r ❡ ❛❣r❛❞❡❝❡r ❛♦ ♠❡✉ ♣❛✐✱ ❏♦sé ❋r❛♥❝✐s❝♦ ❞❡ ❈❛str♦ ✭❡♠ ♠❡♠ór✐❛✮✱ q✉❡ ❞♦ ✐♥t❡r✐♦r ❞❡ P❡r♥❛♠❜✉❝♦ ❝♦♠♦ ❛❣r✐❝✉❧t♦r ❡ ❜ó✐❛ ❢r✐❛ s❡ ♠✉❞♦✉ ♣❛r❛ ❛ ❝❛♣✐t❛❧ tr❛③❡♥❞♦ s♦❜r❡ ♦ ♦♠❜r♦ ♦ tr❛❜❛❧❤♦ ❡ ❛ ❤♦♥❡st✐❞❛❞❡ ♣❛r❛ ♥♦s ❝r✐❛r ❝♦♠ ❞✐❣♥✐❞❛❞❡ ❡ t♦❞♦s t✐✈❡♠♦s ♥♦ss❛ ❡❞✉❝❛çã♦ ❡s❝♦❧❛r ♥❛s ❡s❝♦❧❛s ♣ú❜❧✐❝❛s✳ ❆ss✉♠✐✉ ❡ ❝r✐♦✉ ❛ ❋❛❜✐❛♥❛ ❈❛str♦✱ ♠✐♥❤❛ ✐r♠ã ♠❛✐s ✈❡❧❤❛✱ ❣❡r♦✉ ❡ ❝r✐♦✉ ❛ ❘❡♥❛t❛ ❈❛str♦✱ ♠✐♥❤❛ ✐r♠ã ❞♦ ♠❡✐♦ ❡ ❛ ♠✐♠ q✉❡ ❣❡r♦✉ ❡ ❞✉r❛♥t❡ ✷✾ ❛♥♦s ❡s❝♦❧❤✐ ❝♦♥✈✐✈❡r ❝♦♠ ❡❧❡ ❛té ♠❡ ❝❛s❛r✱ q✉❛♥❞♦ ♠♦❞❡st❛♠❡♥t❡ ♦ ❤♦♥r❡✐ ❡♠ ✈✐❞❛✱ ❛♣r♦✈❡✐t❛♥❞♦ ❛s ♦♣♦rt✉♥✐❞❛❞❡s ♠❡s♠♦ ❝♦♠ ♠✉✐t❛s ❞✐✜❝✉❧❞❛❞❡s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ✜♥❛♥❝❡✐r❛✳ ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ✐♥❞✐s♣❡♥sá✈❡❧ ♠ã❡ ❆♥❛ ▲ú❝✐❛ ❇❛t✐st❛✱ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ♠✐♥❤❛ ❡①✐stê♥❝✐❛✱ ❛ q✉❡♠ ❞❡✈♦ ❝♦♥st❛♥t❡♠❡♥t❡ ♠❡✉ r❡s♣❡✐t♦ ❡ ❛♠♦r✳ ➚s ✐r♠ãs ❞❛ ❝❛r✐❞❛❞❡ q✉❡ ❞✉r❛♥t❡ ✶✵ ❛♥♦s✱ ❡♥tr❡ ♠✐♥❤❛ ✐♥❢â♥❝✐❛ ❡ ❛❞♦❧❡s❝ê♥❝✐❛✱ ❝✉✐✲ ❞❛r❛♠ ❞❡ ♠✐♠ ❡ ♠❡ ♦❢❡r❡❝❡r❛♠ ❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ❡ ❝♦♠♣❧❡♠❡♥t❛r ❝♦♠♦ ❛♦s ♦✉tr♦s ♠❡♥♦r❡s ❝❛r❡♥t❡s ❞❡ ♠✐♥❤❛ é♣♦❝❛ ♥♦ ❊❞✉❝❛♥❞ár✐♦ ▼❛❣❛❧❤ã❡s ❇❛st♦s✱ ♠❛♥t✐❞♦ ♣❡❧❛ ❙❛♥t❛ ❈❛s❛ ❞❡ ♠✐s❡r✐❝ór❞✐❛ ❞♦ ❘❡❝✐❢❡✱ ❣❡r✐❞♦ ♣❡❧❛s ❋✐❧❤❛s ❞❛ ❈❛r✐❞❛❞❡ ❞❡ ❙ã♦ ❱✐❝❡♥t❡ ❞❡ P❛✉❧♦ ❡ ❛té ❤♦❥❡ s✐t✉❛❞♦ ♥♦ ❜❛✐rr♦ ❞❛ ❱ár③❡❛ ❡♠ ❘❡❝✐❢❡✱ ♦♥❞❡ ♥❛s❝✐ ❡ ❢✉✐ ❝r✐❛❞♦✳ ❆❣r❛❞❡ç♦ ❡ ❤♦♠❡♥❛❣❡✐♦ ♠❡✉s ✐r♠ã♦s ❡ ❛♠✐❣♦s P❛✉❧♦ ❙✐❧❛s ❇❛r❜♦s❛✱ ❋á❜✐♦ ❙♦✉t♦ ❞❛ ❙✐❧✈❛✱ ♠❡✉ ♣r✐♠❡✐r♦ ❛❧✉♥♦ ❡ ❤♦❥❡ ❊♥❣❡♥❤❡✐r♦ ❞❡ Pr♦❞✉çã♦✳ ❙♦✉ ❡t❡r♥❛♠❡♥t❡ ❛❣r❛❞❡❝✐❞♦ ❛♦s ❛♠✐❣♦s ❊s❡q✉✐❛s ❆r❛ú❥♦ ❡ ▲✉✐③ ❈❛r❧♦s q✉❡ ♥♦ ❧❡✐t♦ ❞❛ ✐♥❢❡r♠✐❞❛❞❡ ♥♦ ❤♦s♣✐t❛❧ ❞❛s ❝❧í♥✐❝❛s ❞❛ ❯❋P❊ ♠❡ ❛♣♦✐❛r❛♠ ❛té ✜❝❛r ❝✉r❛❞♦ ❡ ♣♦❞❡r✱ ❛ss✐♠✱ ✐♥✐❝✐❛r ♠✐♥❤❛s ❛t✐✈✐❞❛❞❡s ❛❝❛❞ê♠✐❝❛s ♥❛ ❯❋❘P❊✱ ♥❛ q✉❛❧ r❡❝❡❜✐ ♠❡✉ ❣r❛✉ ❞❡ ❧✐❝❡♥❝✐❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛✳ ❆❣r❛❞❡ç♦ ✈❡❡♠❡♥t❡♠❡♥t❡ ❛ ♠✐♥❤❛ q✉❡r✐❞❛ ❡s♣♦s❛ ▼❛r❝❡❧❛ ❈❛str♦ q✉❡ ♠✉✐t❛s ✈❡③❡s ♥❛s ❝❛❧❛❞❛s ❞❛ ♥♦✐t❡ t❡st❡♠✉♥❤❛✈❛ t❛❧ s❛❝r✐❢í❝✐♦ ❛té ❛ ❛✉r♦r❛ ❞♦ ❞✐❛ ❡ ❡♥t❡♥❞❡✉ t❛♠❛♥❤♦ ❡s❢♦rç♦✳ ❚❛♠❜é♠ s♦✉ ❣r❛t♦ ❛♦ ♠❡✉ ✜❧❤♦ ❉❛✈✐ ❋❛❜✐❛♥♦ q✉❡ ♠❡s♠♦ ♥❛ s✉❛ t❡♥r❛ ✐❞❛❞❡✱ ♠✉✐t❛s ✈❡③❡s ❝♦♠♣r❡❡♥❞❡✉ ♦ ❡♥❢❛❞♦ ❡ ❝❛♥ç❛s♦ ❢ís✐❝♦✲♠❡♥t❛❧ ❞❡ s❡✉ ♣❛♣❛✐✳ ❙♦✉ ♠✉✐t♦ ❛❣r❛❞❡❝✐❞♦ t❛♠❜é♠ ❛♦s ❉♦✉t♦r❡s Pr♦❢❡ss♦r❡s ◆❛♣♦❧❡♦♥ ❈❛r♦ ❚✉❡st❛✱ ▲✐③❛♥❞r♦ ❈❤❛❧❧❛♣❛✱ ❆❧❡①❛♥❞r❡ ❙✐♠❛s✱ ❇r✉♥♦ ❘✐❜❡✐r♦✱ ▲❡♥✐♠❛r ◆✉♥❡s✱ ❙ér❣✐♦ ❞❡ ❆❧❜✉q✉❡rq✉❡✱ ●✐❧♠❛r ❈♦rr❡✐❛ ❡ ❊❞✉❛r❞♦ ●♦♥ç❛❧✈❡s✱ ❞♦❝❡♥t❡s ✐❧✉♠✐♥❛❞♦s ❞❡ss❡ ♣r♦❣r❛♠❛✱ q✉❡ ❞✉r❛♥t❡ t♦❞❛ ❛ ♠✐♥❤❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛✱ ❝♦♠♦ ♠❡str❛♥❞♦ ♥♦ ❡st❛❞♦ ❞❛ P❛r❛í❜❛✱ ❝♦♠♣❛rt✐❧❤❛r❛♠ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ s❡♠ ❤❡s✐t❛r ❞❡❞✐❝❛r❛♠ s❡✉ t❡♠♣♦ ♥❛ ❝♦♥str✉çã♦ ♣r♦✜ss✐♦♥❛❧ ❞❡ ✉♠ ❡❞✉❝❛❞♦r✳ ❉❡st❛❝♦ ♠✐♥❤❛ ❣r❛t✐❞ã♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢❡ss♦r ❉♦✉t♦r ❈❛r❧♦s ❇♦❝❦❡r ◆❡t♦✱ ♣♦r ♠❡ ♦r✐❡♥t❛r ❝♦♠ ❞❡❞✐❝❛çã♦✱ ❛ss✐❞✉✐❞❛❞❡ ❡
❉❡❞✐❝❛tór✐❛
❆♦ ♠❡✉ ❣❡♥❡r♦s♦ P❛✐ ✭❡♠ ♠❡♠ór✐❛✮✱ ▼ã❡✱ ❊s♣♦s❛ ❡ ❋✐❧❤♦✳
❘❡s✉♠♦
▼♦str❛r❡✐ ♥❡st❛ ❞✐ss❡rt❛çã♦ ❛s r❡❝♦rrê♥❝✐❛s ❧✐♥❡❛r❡s ❝♦♠❡ç❛♥❞♦ ❝♦♠ ✉♠ ❜r❡✈❡ ❝♦♠❡♥tár✐♦ ❤✐stór✐❝♦ s♦❜r❡ ♦s ♣r✐♥❝✐♣❛✐s ❛✉t♦r❡s ❞❡ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❞❡ r❡❝♦rrê♥❝✐❛s ❧✐♥❡❛r❡s✳ ❆♥❛❧✐s❛r❡✐ s❡q✉ê♥❝✐❛s ❡❧❡♠❡♥t❛r❡s✱ ❢ór♠✉❧❛s ♣♦s✐❝✐♦♥❛✐s✱ ♠ét♦❞♦s r❡❝✉rs✐✲ ✈♦s✱ ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ❡ ❣❡♦♠étr✐❝❛s✳ P♦st❡r✐♦r♠❡♥t❡✱ ❞✐❢❡r❡♥❝✐❛r❡✐ ♦ q✉❡ sã♦ r❡❧❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s ❡ ❡q✉❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s s❡❣✉✐♥❞♦ ❝♦♠ ❛ ❡①♣❧✐❝❛çã♦ ❞❛ s♦❧✉çã♦ ❞❡ ✉♠❛ r❡❝♦rrê♥❝✐❛✱ ❡①♣♦st❛ ❡♠ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❡ t❛♠❜é♠ ❛s ❞❡✜♥✐çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s ❞❡ ♣r✐♠❡✐r❛ ❡ s❡❣✉♥❞❛ ♦r❞❡♥s ❝♦♠ s✉❛s ❝❧❛ss✐✜❝❛çõ❡s✳ ▲♦❣♦ ❛♣ós ❞✐s✲ ❝♦rr❡r❡✐✱ ❜r❡✈❡♠❡♥t❡✱ à r❡s♣❡✐t♦ ❞❡ ❛❧❣✉♥s t✐♣♦s ❞❡ r❡❝♦rrê♥❝✐❛s ❞❡ t❡r❝❡✐r❛ ♦r❞❡♠ ❡ ✈❡r❡♠♦s t❛♠❜é♠ ❛❧❣✉♠❛s ❣❡♥❡r❛❧✐③❛çõ❡s ♣❛r❛ ♦r❞❡♠ s✉♣❡r✐♦r✳ ❚r❛t❛r❡✐✱ ✜♥❛❧✐③❛♥❞♦ ♥❡st❡ tr❛❜❛❧❤♦✱ ❛♣❧✐❝❛çõ❡s ❞❛s r❡❝♦rrê♥❝✐❛s ✉t✐❧✐③❛♥❞♦ ❛s ❢✉♥❞❛♠❡♥t❛çõ❡s r❡❢❡r✐❞❛s ❛♥t❡r✐♦r♠❡♥t❡ ❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ❝♦♠❜✐♥❛tór✐❛✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛✱ ❊♥s✐♥♦ ❇ás✐❝♦✱ ❙❡q✉ê♥❝✐❛s✱ ❘❡❝♦rrê♥✲ ❝✐❛s✱ ❆♣❧✐❝❛çõ❡s✳
❆❜str❛❝t
■ s❤♦✇ t❤✐s t❤❡s✐s ❧✐♥❡❛r r❡❝✉rr❡♥❝❡s st❛rt✐♥❣ ✇✐t❤ ❛ ❜r✐❡❢ ❤✐st♦r✐❝❛❧ r❡✈✐❡✇ ♦❢ t❤❡ ♠❛✐♥ ❛✉t❤♦rs ♦❢ s♦♠❡ ♣r♦❜❧❡♠s ♦❢ ❧✐♥❡❛r r❡❝✉rr❡♥❝❡s✳ ❆♥❛❧②③❡ ❡❧❡♠❡♥t❛r② s❡q✉❡♥✲ ❝❡s✱ ♣♦s✐t✐♦♥❛❧ ❢♦r♠✉❧❛s✱ r❡❝✉rs✐✈❡ ♠❡t❤♦❞s✱ ❛r✐t❤♠❡t✐❝ ❛♥❞ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥s✳ ▲❛t❡r✱ ■ ✇✐❧❧ ❞✐st✐♥❣✉✐s❤ ✇❤❛t ❛r❡ r❡❧❛t✐♦♥s ♦❢ r❡❧❛♣s❡s ❛♥❞ r❡❝✉rr❡♥❝❡s ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s ✇✐t❤ t❤❡ ❡①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛ r❡❝✉rr❡♥❝❡✱ ❡①♣♦s❡❞ ✐♥ s♦♠❡ ✐♥s✲ t❛♥❝❡s ❛♥❞ ❛❧s♦ t❤❡ ✜rst r❡❝✉rr❡♥❝❡ s❡tt✐♥❣s ❛♥❞ s❡❝♦♥❞ ♦r❞❡rs ✇✐t❤ t❤❡✐r r❛t✐♥❣s✳ ❙♦♦♥ ❛❢t❡r ■✬❧❧ ❞✐s❝✉ss ❜r✐❡✢② t❤❡ r❡s♣❡❝t ♦❢ s♦♠❡ t②♣❡s ♦❢ t❤✐r❞✲♦r❞❡r r❡❝✉rr❡♥❝❡s ❛♥❞ ❛❧s♦ s❡❡ s♦♠❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s t♦ ❤✐❣❤❡r ♦r❞❡r✳ ❚r❡❛t✱ ✜♥✐s❤✐♥❣ t❤✐s ✇♦r❦✱ ❛♣♣❧✐❝❛t✐✲ ♦♥s ♦❢ r❡❝✉rr❡♥❝❡s ✉s✐♥❣ t❤❡ ❢♦✉♥❞❛t✐♦♥s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡ ❛♥❞ ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ ❝♦♠❜✐♥❛t♦r✐❛❧✳
❑❡②✇♦r❞s✿ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✱ ❇❛s✐❝ ❊❞✉❝❛t✐♦♥✱ ❙❡q✉❡♥❝❡s✱ r❡❝✉rr❡♥❝❡s✱ ❆♣♣❧✐❝❛t✐♦♥s✳
❙✉♠ár✐♦
✶ ❙❡q✉ê♥❝✐❛s ❊❧❡♠❡♥t❛r❡s ✶
✶✳✶ ❯♠ ❜r❡✈❡ ❝♦♠❡♥tár✐♦ ❤✐stór✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❋ór♠✉❧❛s P♦s✐❝✐♦♥❛✐s ❡ ▼ét♦❞♦s ❘❡❝✉rs✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ Pr♦❣r❡ssõ❡s ❆r✐t♠ét✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹ Pr♦❣r❡ssõ❡s ●❡♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷ ❘❡❝♦rrê♥❝✐❛s ✶✹
✷✳✶ ❘❡❧❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s✱ ❡q✉❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ❈❧❛ss✐✜❝❛çã♦ ❞❡ ❙❡q✉ê♥❝✐❛s ❘❡❝♦rr❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✸ ❙♦❧✉çã♦ ❞❡ ✉♠❛ ❘❡❝♦rrê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✹ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ Pr✐♠❡✐r❛ ❖r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✹✳✶ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ Pr✐♠❡✐r❛ ❖r❞❡♠ ❍♦♠♦❣ê♥❡❛s ✳ ✳ ✳ ✳ ✶✽ ✷✳✹✳✷ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ Pr✐♠❡✐r❛ ❖r❞❡♠ ◆ã♦✲❍♦♠♦❣ê♥❡❛s ✳ ✳ ✶✾ ✷✳✺ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✺✳✶ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ ❍♦♠♦❣ê♥❡❛s ✳ ✳ ✳ ✳ ✷✷ ✷✳✺✳✷ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠
◆ã♦✲❍♦♠♦❣ê♥❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✻ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ ❖r❞❡♠ ❙✉♣❡r✐♦r ✲ ❯♠❛ ●❡♥❡r❛❧✐③❛çã♦ ✳ ✳ ✳ ✳ ✸✵ ✷✳✻✳✶ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ ♦r❞❡♠ n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✷✳✻✳✷ ❈❛s♦s P❛rt✐❝✉❧❛r❡s ❞❡ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ ❚❡r❝❡✐r❛ ❖r✲ ❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✸ ❆♣❧✐❝❛çõ❡s ❞❡ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s ✸✺
✸✳✶ ❆ ❚♦rr❡ ❞❡ ❍❛♥♦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ ❆ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷✳✶ ❖ ❈á❧❝✉❧♦ ❞♦ ❚❛♠❛♥❤♦ ❞❡ ❯♠❛ P♦♣✉❧❛çã♦ ❞❡ ❈♦❡❧❤♦s ✳ ✳ ✳ ✳ ✹✵ ✸✳✷✳✷ ❆s P❡ç❛s ❞❡ ❯♠❛ ❈❛✐①❛ ❞❡ ❉♦♠✐♥ó ✷ ① ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✸ ❖ Pr♦❜❧❡♠❛ ❞♦s ❈❛♠✐♥❤♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✹ ❖s ◆ú♠❡r♦s ❞❡ ❙t✐r❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✹✳✶ ◆ú♠❡r♦ ❞❡ ❙t✐r❧✐♥❣ ❞❡ ❙❡❣✉♥❞♦ ❚✐♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✹✳✷ ◆ú♠❡r♦ ❞❡ ❙t✐r❧✐♥❣ ❞❡ Pr✐♠❡✐r♦ ❚✐♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✸✳✺ ❖ Pr♦❜❧❡♠❛ ❞❡ ❏♦s❡♣❤✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❉r❛✇✐♥❣ ❍❛♥❞s ❞❡ ▼✳❈✳❊s❝❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❈♦♥❥✉♥t♦ ❋r❛❝t❛✐s ▼❛♥❞❡❧❜r♦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✸✳✶ ❚♦rr❡ ❞❡ ❍❛♥♦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ ❚❡r❡♠♦s ❛♣❡♥❛s ✉♠ ♠♦✈✐♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✸ ❚❡r❡♠♦s três ♠♦✈✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✹ ❚❡r❡♠♦s ❡①❛t❛♠❡♥t❡ s❡t❡ ♠♦✈✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✺ ❚♦rr❡ ❝♦♠ n ❞✐s❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✻ ❚♦rr❡ ❝♦♠ n−1 ❞✐s❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✼ ❚♦rr❡ ❝♦♠ ♦ ♠❛✐♦r ❞✐s❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✽ ❚♦rr❡ ❝♦♠ n−1 ❞✐s❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✾ ❋✐❜♦♥❛❝❝✐ ▲✐❜❡r ❆❜❛❝✐ ❇♦♦❦ ❖❢ ❈❛❧❝✉❧❛t✐♦♥ ❆✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✶✵ ➪r✈♦r❡ ❣❡♥❡❛❧ó❣✐❝❛ ❛té ♦ q✉❛rt♦ ♠ês ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✶✶ ➪r✈♦r❡ ❣❡♥❡❛❧ó❣✐❝❛ ❛té ♦ sét✐♠♦ ♠ês ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✶✷ ❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✶✸ ❈❛✐①❛ ❉♦♠✐♥ó 2×n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✸✳✶✹ P♦ss✐❜✐❧✐❞❛❞❡ ❝♦♠ ✉♠❛ ♣❡ç❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✶✺ P♦ss✐❜✐❧✐❞❛❞❡s ❝♦♠ ❞✉❛s ♣❡ç❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✶✻ P♦ss✐❜✐❧✐❞❛❞❡s ❝♦♠ três ♣❡ç❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✶✼ P♦ss✐❜✐❧✐❞❛❞❡s ❝♦♠ q✉❛tr♦ ♣❡ç❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✶✽ ❈❛♠✐♥❤♦s ♣❡❧♦s ❙❡❣♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✶✾ ❈❛♠✐♥❤♦s ❖r✐❡♥t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✷✵ ❯♠ ❈❛♠✐♥❤♦ P♦ssí✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✷✶ ❉♦✐s ❈❛♠✐♥❤♦s P♦ssí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✷✷ ◗✉❛tr♦ ❈❛♠✐♥❤♦s P♦ssí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✷✸ ❙❡t❡ ❈❛♠✐♥❤♦s P♦ssí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✷✹ ❚r❡③❡ ❈❛♠✐♥❤♦s P♦ssí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✷✺ ●❡♥❡r❛❧✐③❛♥❞♦ P♦ssí✈❡✐s ❈❛♠✐♥❤♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✷✻ ❉✐s♣♦s✐çã♦ ❞♦s ♣r✐s✐♦♥❡✐r♦s ❡♠ ❞✉❛s ♠❡s❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✷✼ ❊❧✐♠✐♥❛çã♦ ♣♦r ✈♦❧t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✷✽ ❚❛❜❡❧❛ ❞❡ ❡❧✐♠✐♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✷✾ ❊❧✐♠✐♥❛çã♦ ♣❛r❛ ♦ ❝❛s♦ ✷♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
✸✳✸✵ ❊❧✐♠✐♥❛çã♦ ♣❛r❛ ♦ ❝❛s♦ ✷♥✰✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✸✳✸✶ ❚❛❜❡❧❛ ❞❡ ❡❧✐♠✐♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
■♥tr♦❞✉çã♦
◆♦ q✉❡ t❛♥❣❡ à ✐♠♣♦rtâ♥❝✐❛ ❞♦ t❡♠❛✱ ❛s s❡q✉ê♥❝✐❛s sã♦ ❢✉♥❞❛♠❡♥t❛✐s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ à ❞❡s♣❡✐t♦ ❞❡ s❡r ♠✉✐t♦ ❧✐♠✐t❛❞♦ ♦ ❡♥s✐♥♦ ❞❡ r❡❝♦rrê♥❝✐❛s ♥❛s sér✐❡s ✜♥❛✐s ❡ ❡♥s✐♥♦ ♠é❞✐♦✱ ❛❝r❡❞✐t❛♠♦s q✉❡ ❡st❡ ❛ss✉♥t♦ ♣r❡❝✐s❛ s❡r ♠❛✐s ❞✐❢✉♥❞✐❞♦ ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦ ❡ s✉♣❡r✐♦r✳
P♦✉❝♦ é ❞✐s❝✉t✐❞♦ ♦✉ ♣r❛t✐❝❛♠❡♥t❡ ♥❛❞❛ s♦❜r❡ r❡❧❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s ❡ ❡q✉❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s✱ ❛♣❡♥❛s ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛ ❡ ❣❡♦♠étr✐❝❛✳ ❆❧❣✉♥s t✐♣♦s ❞❡ s❡q✉ê♥✲ ❝✐❛s ❡ ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s ❞❡ ❞✐❢í❝✐❧ r❡s♦❧✉çã♦ ♣♦❞❡r✐❛♠ ❛♣♦✐❛r s✉❛ r❡s♦❧✉çã♦ ♥♦ ♠ét♦❞♦ r❡❝✉rs✐✈♦✳ ➱ ♣❡♥s❛♥❞♦ ❛ss✐♠ q✉❡ ❡ss❡ ♠ét♦❞♦ s❡ t♦r♥❛ ✐♠♣r❡s❝✐♥❞í✈❡❧ ❞✐❛♥t❡ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝r✐❛r ❢ór♠✉❧❛s q✉❡ r❡s♦❧✈❡♠ ♣❛rt❡s ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❣❡♠ ❡ ✐t❡r❛çõ❡s ❛♣❧✐❝❛❞❛s ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♥❛s ❞❡❞✉çõ❡s ❞❡ ❢ór♠✉❧❛s ❡ ❛ ❞❡s❝♦❜❡rt❛ ❞❡ ✈ár✐♦s ♠ét♦❞♦s ♦♥❞❡ ❡♥❝♦♥tr❛♠♦s r❡❝✉rs♦s ❞✐❞át✐❝♦s ♣❛r❛ r❡s♦❧✉çã♦ ❝♦♠ ❝❧❛r❡③❛✳
❆❝r❡❞✐t❛♠♦s✱ ❝♦♠♦ ❥á ❢❛❧❛♠♦s ❛♥t❡s✱ s❡r ✐♠♣♦rt❛♥t❡ ❡st✉❞❛r ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦ ❡st❡ t❡♠❛ r✐❝♦ ❡ ❝♦♥t✐♥✉❛r ♥♦ ❡♥s✐♥♦ ❞❡ ❣r❛❞✉❛çã♦ ♣❛r❛ ❢❛❝✐❧✐t❛r ❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❛❧❣✉♠❛s ❢ór♠✉❧❛s✱ ✈✐st♦ ♥♦s ❛ss✉♥t♦s ❝♦♠♦✱ s❡q✉ê♥❝✐❛s✱ ❛s ♠❛✐s ❞✐✈❡rs❛s✱ ♣r♦❣r❡ssõ❡s ❡ ♣r♦❜❧❡♠❛s ❝♦♠❜✐♥❛tór✐♦s✱ ❜❡♠ ❝♦♠♦ ♥❛s ❛♣❧✐❝❛çõ❡s ❡♠ ♣r♦❜❧❡♠❛s ❝♦♠♣❧❡①♦s ❞❡ ❞✐❢í❝✐❧ r❡s♦❧✉çã♦✳
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ♠♦str❛r❡♠♦s ❛❧❣✉♥s t✐♣♦s ❡❧❡♠❡♥t❛r❡s ❞❡ s❡q✉ê♥❝✐❛s ❝♦♠ s✉❛s ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ✈á❧✐❞❛s ♣❛r❛ s❡q✉ê♥❝✐❛s ❡♠ ❣❡r❛❧✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠ ❜r❡✈❡ ❝♦♠❡♥tár✐♦ ❤✐stór✐❝♦ s♦❜r❡ r❡❝♦rrê♥❝✐❛s ❧✐♥❡❛r❡s ❝♦♠ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ❝r✐❛❞♦r❡s✱ tr❛t❛r❡♠♦s ❞❡ r❡❧❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s✱ ❡q✉❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s ❡ ❝❧❛ss✐✜❝❛r❡♠♦s ❛s r❡❧❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s ❞❡st❛❝❛♥❞♦ s✉❛s ♦r❞❡♥s✱ s❡ ❤♦♠♦❣ê♥❡❛ ♦✉ ♥ã♦✳
❆♠♣❧✐❛r❡♠♦s ❡♠ s❡❣✉✐❞❛ ❛ ❞✐s❝✉ssã♦ s♦❜r❡ r❡❝♦rrê♥❝✐❛s ❧✐♥❡❛r❡s ❞❡ ♦r❞❡♥s ♠ú❧t✐✲ ♣❧❛s✱ ♥♦ q✉❡ t❛♥❣❡ ❛♦ ❡st✉❞♦ ❞❡ ❛❧❣✉♥s t✐♣♦s ❞❡ r❡❝♦rrê♥❝✐❛s ❞❡ ♦r❞❡♠ 3✱ ❞❡st❛❝❛♥❞♦
t❛♠❜é♠ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ r❡❧❛❝✐♦♥❛♥❞♦ ❝♦♠ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❡♠ r❡❝♦rrê♥❝✐❛s ❞❡ ♦r❞❡♠ k✳
◆♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ❞❡✐①❛r❡♠♦s ❡s♣❡❝✐❛❧♠❡♥t❡ ♣❛r❛ ❛s ❛♣❧✐❝❛çõ❡s ❞❛s r❡❝♦rrê♥❝✐❛s ❧✐♥❡❛r❡s✱ ❡♠ ❞❡st❛q✉❡ ❛♦s três ❢❛♠♦s♦s ♣r♦❜❧❡♠❛s✳ ❆s ❛♣❧✐❝❛çõ❡s ❞❛ t♦rr❡ ❞❡ ❍❛♥♦✐ ✲ ➱❞♦✉❛r❞ ▲✉❝❛s✱ ❞❛s ♠✉❧t✐♣❧✐❝❛çõ❡s ❞♦s ❝♦❡❧❤♦s ✲ ❙ér✐❡s ❞❡ ❋✐❜♦♥❛❝❝✐ ❡ ❞♦ s❛❧✈❛✲ ♠❡♥t♦ ❡♠ ✉♠❛ ❡♠❜❛r❝❛çã♦ ✲ ❋❧❛✈✐✉s ❏♦s❡♣❤✉s✳ ▼♦str❛r❡♠♦s t❛♠❜é♠ ♥❛ ❛♣r❡s❡♥✲ t❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦ q✉❡ é ♣♦ssí✈❡❧ ✐♥♦✈❛r ♣❡❞❛❣♦❣✐❝❛♠❡♥t❡ ♦ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠
❞❡ t❡♠❛s r❡❧❛❝✐♦♥❛❞♦s ❛ ❘❡❝♦rrê♥❝✐❛s✱ ❢❛③❡♥❞♦ ✉s♦ ❞❡ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s✱ r❡❝✉r✲ s♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❞❡ r♦❜ót✐❝❛✱ ❝♦♠♦ ♥♦✈❛s ❢❡rr❛♠❡♥t❛s t❡❝♥♦❧ó❣✐❝❛s ♥❛ ❡s❝♦❧❛ ♦✉ ✉♥✐✈❡rs✐❞❛❞❡✳
❊♥✜♠✱ é ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ r❡❢♦rç❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ t❛♥t♦ ♥❛ ♣rát✐❝❛ ❝♦♠♦ ♥❛ t❡♦r✐❛✱ s♦❜r❡ ❡ss❡ ❛ss✉♥t♦ tã♦ ❡♥✈♦❧✈❡♥t❡ ♥❛ ♠❛t❡♠át✐❝❛ ❞✐s❝r❡t❛ ❡ ♣♦r ✐ss♦ tr❛✲ t❛r❡♠♦s ❞❡ ♠♦str❛r ❝♦♠ ♠✉✐t♦ ❛✜♥❝♦ ♥❡st❡ ❡s❢♦rç❛❞♦ tr❛❜❛❧❤♦ q✉❡ ♥♦ ❡♥s❡❥♦ ❛♣r❡✲ s❡♥t❛♠♦s✳
❈❛♣ít✉❧♦ ✶
❙❡q✉ê♥❝✐❛s ❊❧❡♠❡♥t❛r❡s
✶✳✶
❯♠ ❜r❡✈❡ ❝♦♠❡♥tár✐♦ ❤✐stór✐❝♦
▼✉✐t❛s s❡q✉ê♥❝✐❛s ✐♠♣♦rt❛♥t❡s sã♦ ❞❡✜♥✐❞❛s r❡❝✉rs✐✈❛♠❡♥t❡✱ ❢♦r♥❡❝❡♥❞♦✲s❡ ✐♥✐✲ ❝✐❛❧♠❡♥t❡ ✉♠❛ ❢ór♠✉❧❛ q✉❡ ❞❡t❡r♠✐♥❛ ♦s ❞❡♠❛✐s t❡r♠♦s ❛ ♣❛rt✐r ❞♦s t❡r♠♦s q✉❡ ♦s ♣r❡❝❡❞❡♠✳ ❊ss❛ ❢ór♠✉❧❛ é ❝❤❛♠❛❞❛ ❞❡ r❡❝♦rrê♥❝✐❛✳ ❆ ❢♦r♠✉❧❛çã♦ ❞❡ r❡❧❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s é ✉♠ r❡❝✉rs♦ ❢♦rt❡ ❡ ✈❡rsát✐❧ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❝♦♠❜✐♥❛tór✐♦s ❡ ♠✉✐t♦s ❛❧❣♦r✐t♠♦s sã♦ ❜❛s❡❛❞♦s ❡♠ r❡❧❛çõ❡s r❡❝♦rr❡♥t❡s✱ ♣❛rt❡ ❞❡st❡s ♣r♦❜❧❡♠❛s ❝♦♥s✐❞❡r❛❞♦s ❞✐❢í❝❡✐s à ♣r✐♠❡✐r❛ ♠ã♦ sã♦ ❝❧❛r❛♠❡♥t❡ r❡s♦❧✈✐❞♦s ♣♦r ❡st❛ té❝♥✐❝❛✳ ❊st❡ t❡♠❛ é ♠✉✐t♦ út✐❧ t❛♠❜é♠ ❡♠ ♣❡sq✉✐s❛s ❡ ♥❛ ✈✐❞❛✱ ❝♦♠♦ ❛ r❡❝✉rsã♦ q✉❡ ♦❝♦rr❡ ♥♦ ❞♦♠í♥✐♦ ✈✐s✉❛❧✱ ❞♦ tr❛❜❛❧❤♦ ❞❡ ▼✳❈✳❊s❝❤❡r ❡ t❛♠❜é♠ ♣r❡s❡♥t❡ ♥♦s ❢r❛❝t❛✐s ♣r❡s❡♥❝✐✲ ❛❞♦s ♣♦r ▼❛♥❞❡❧❜r♦t ❝♦♠♦ ♠♦str❛♠ ❛s ✜❣✉r❛s ❛❜❛✐①♦✳ ✭ ❱❡❥❛ ❛s ✜❣✉r❛s ❡♠✿ ❉r❛✇✐♥❣ ❍❛♥❞s ✲ ❊s❝❤❡r ▼✳❈✳ ✲ ❲✐❦✐❆rt✳♦r❣ ✇✇✇✳✇✐❦✐❛rt✳♦r❣ ❉r❛✇✐♥❣ ❍❛♥❞s ✲ ❊s❝❤❡r ▼✳❈✳ ❡ ❖s ❋r❛❝t❛✐s ⑤ ❚❡♦r✐❛ ❞❛ ❈♦♥s♣✐r❛çã♦ ✇✇✇✳❞❡❧❞❡❜❜✐♦✳❝♦♠✳❜r✮
❋✐❣✉r❛ ✶✳✶✿ ❉r❛✇✐♥❣ ❍❛♥❞s ❞❡
▼✳❈✳❊s❝❤❡r ❋✐❣✉r❛ ✶✳✷✿ ❈♦♥❥✉♥t♦ ❋r❛❝t❛✐s ▼❛♥❞❡❧❜r♦t
✶✳✶✳ ❯▼ ❇❘❊❱❊ ❈❖▼❊◆❚➪❘■❖ ❍■❙❚Ó❘■❈❖
❊♠ s❡ tr❛t❛♥❞♦ ❞❡st❡ ✐♠♣♦rt❛♥t❡ r❡❝✉rs♦✱ ❞❡st❛❝❛r❡♠♦s três ❢❛♠♦s❛s ❛♣❧✐❝❛çõ❡s✱ ❞❡♥tr❡ ❛s ❞❡♠❛✐s q✉❡ ❡st✉❞❛r❡♠♦s ♠❛✐s ❛❞✐❛♥t❡✱ ❝♦♠ s❡✉s r❡s♣❡❝t✐✈♦s ❛✉t♦r❡s✳
◆❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❘❡❝♦rrê♥❝✐❛s ▲✐♥❡❛r❡s t❡♠♦s ❛ ❢❛♠♦s❛ t♦rr❡ ❞❡ ❍❛♥♦✐✱ ❡st❡ ❥♦❣♦ ❢♦✐ ✐♥✈❡♥t❛❞♦ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ➱❞♦✉❛r❞ ▲✉❝❛s(1842−1891)✐♥s♣✐r❛❞♦
♥✉♠❛ ❧❡♥❞❛ ❍✐♥❞✉✱ ❡♠ 1883✳ ❖ ♥♦♠❡ ❞♦ ❥♦❣♦ s✉r❣✐✉ ❞♦ sí♠❜♦❧♦ ❞❛ ❝✐❞❛❞❡ ❞❡ ❍❛♥♦✐✱
♥♦ ❱✐❡t♥ã✳ ❊①✐st❡♠ ✈ár✐❛s ❧❡♥❞❛s ❛ r❡s♣❡✐t♦ ❞❛ ♦r✐❣❡♠ ❞♦ ❥♦❣♦✱ ❛ ♠❛✐s ❝♦♥❤❡❝✐❞❛ ❞✐③ r❡s♣❡✐t♦ ❛ ✉♠ t❡♠♣❧♦ ❇❡♥❛r❡s✱ s✐t✉❛❞♦ ♥♦ ❝❡♥tr♦ ❞♦ ❯♥✐✈❡rs♦✳ ❉✐③✲s❡ q✉❡ ✉♠ s❡r s✉♣❡r✐♦r✱ s✉♣♦st❛♠❡♥t❡✱ ❤❛✈✐❛ ❝r✐❛❞♦ ✉♠❛ t♦rr❡ ❝♦♠ 64 ❞✐s❝♦s ❞❡ ♦✉r♦ ❡ ♠❛✐s ❞✉❛s
❡st❛❝❛s ❡q✉✐❧✐❜r❛❞❛s s♦❜r❡ ✉♠❛ ♣❧❛t❛❢♦r♠❛✳ ❊ss❛ ❞✐✈✐♥❞❛❞❡ ♦r❞❡♥❛r❛ ❛♦s ♠♦♥❣❡s q✉❡ ♠♦✈❡ss❡♠ t♦❞♦s ♦s ❞✐s❝♦s ❞❡ ✉♠❛ ❡st❛❝❛ ♣❛r❛ ♦✉tr❛ s❡❣✉♥❞♦ às s✉❛s ✐♥str✉çõ❡s✳ ❆s r❡❣r❛s ❡r❛♠ s✐♠♣❧❡s✿ ❛♣❡♥❛s ✉♠ ❞✐s❝♦ ♣♦❞✐❛ s❡r ♠♦✈✐❞♦ ❞❡ ❝❛❞❛ ✈❡③ ❡ ♥✉♥❝❛ ✉♠ ❞✐s❝♦ ♠❛✐♦r ❞❡✈❡r✐❛ ✜❝❛r ♣♦r ❝✐♠❛ ❞❡ ✉♠ ❞✐s❝♦ ♠❡♥♦r✳ ❙❡❣✉♥❞♦ ❛ ❧❡♥❞❛✱ q✉❛♥❞♦ t♦❞♦s ♦s ❞✐s❝♦s ❢♦ss❡♠ tr❛♥s❢❡r✐❞♦s ❞❡ ✉♠❛ ❡st❛❝❛ ♣❛r❛ ❛ ♦✉tr❛✱ ♦ t❡♠♣❧♦ ❞❡s♠♦r♦♥❛r✲s❡✲✐❛ ❡ ♦ ♠✉♥❞♦ ❞❡s❛♣❛r❡❝❡r✐❛✳ ❉❡ss❛ ❢♦r♠❛ ❝r✐❛✈❛✲s❡ ✉♠ ♥♦✈♦ ♠✉♥❞♦✱ ♦ ♠✉♥❞♦ ❞❡ ❍❛♥♦✐✳ ❊ss❡ ❝♦♠❡♥tár✐♦ ❤✐stór✐❝♦ ♣♦❞❡♠♦s ✈❡r ❝♦♠ ♠❛✐s ❞❡t❛❧❤❡s ♥❛ r❡❢❡rê♥❝✐❛ ❬✶✵❪✳
❋✐❜♦♥❛❝❝✐✱ ✜❧❤♦ ❞❡ ❇♦♥❛❝❝✐♦✱ (1175−1250)✱ s❡❣✉♥❞♦ ❛❧❣✉♥s ❡st✉❞✐♦s♦s✱ ❢♦✐ ♦
♠❛t❡♠át✐❝♦ ♠❛✐s t❛❧❡♥t♦s♦ ❞❛ ■❞❛❞❡ ▼é❞✐❛✳ ◆❛t✉r❛❧ ❞❡ P✐s❛✱ ■tá❧✐❛✱ ❡r❛ t❛♠❜é♠ ❝♦✲ ♥❤❡❝✐❞♦ ❝♦♠♦ ▲❡♦♥❛r❞♦ ❞❡ ♣✐s❛ ♦✉ ▲❡♦♥❛r❞♦ P✐s❛♥♦✳ ❆s ❛t✐✈✐❞❛❞❡s ♠❡r❝❛♥t✐s ❞❡ s❡✉ ♣❛✐✱ ❛❧é♠ ❞❛ ♣ró♣r✐❛ ✈♦❝❛çã♦ ❝♦♠❡r❝✐❛❧ ❞❛ ❝✐❞❛❞❡✱ ❢❛✈♦r❡❝❡r❛♠ ❛ ▲❡♦♥❛r❞♦ ❛ ♦♣♦rt✉✲ ♥✐❞❛❞❡ ❞❡ ❡st✉❞❛r ❢♦r❛ ❞❛ ■tá❧✐❛ ❡ ❞❡ ✈✐❛❥❛r ❡♥tr❛♥❞♦ ❡♠ ❝♦♥t❛t♦ ❝♦♠ ♦ ♣❡♥s❛♠❡♥t♦ ♠❛t❡♠át✐❝♦ ár❛❜❡ ❡ ❞♦ ♦r✐❡♥t❡✳ ❊♠ s❡✉ ❧✐✈r♦ ▲✐❜❡r ❆❜❛❝✐✱ ♣✉❜❧✐❝❛❞♦ ❡♠ 1202 ❛ss✐♠
q✉❡ r❡t♦r♥♦✉ ❞❡ s✉❛s ✈✐❛❣❡♥s✱ ❋✐❜♦♥❛❝❝✐ ❞❡❢❡♥❞❡✉ ❝♦♠ ✈✐❣♦r ❛ ❛❞♦çã♦ ❞♦ s✐st❡♠❛ ❞❡ ♥✉♠❡r❛çã♦ ✐♥❞♦✲❛rá❜✐❝♦ ❡♠ ❧✉❣❛r ❞♦s ❛❧❣❛r✐s♠♦s r♦♠❛♥♦s ❡♥tã♦ ✉t✐❧✐③❛❞♦s✳ ➱ ♥❡st❡ ♠❡s♠♦ ❧✐✈r♦ q✉❡ ❡♥❝♦♥tr❛♠♦s✱ ❡♥tr❡ ♦✉tr♦s✱ ♦ ♣r♦❜❧❡♠❛ q✉❡ ❞❡✉ ♦r✐❣❡♠ à ❢❛♠♦s❛ s❡q✉ê♥❝✐❛✳ ◗✉❛♥t♦s ♣❛r❡s ❞❡ ❝♦❡❧❤♦s s❡rã♦ ♣r♦❞✉③✐❞♦s ♥✉♠ ❛♥♦✱ à ♣❛rt✐r ❞❡ ✉♠ ú♥✐❝♦ ❝❛s❛❧✱ s❡ ❝❛❞❛ ❝❛s❛❧ ♣r♦❝r✐❛ ❛ ❝❛❞❛ ♠ês ✉♠ ♥♦✈♦ ❝❛s❛❧ q✉❡ s❡ t♦r♥❛ ♣r♦❞✉t✐✈♦ ❞❡♣♦✐s ❞❡ ❞♦✐s ♠❡s❡s❄ ❆ q✉❡stã♦ é ❢❛s❝✐♥❛♥t❡ t❛♥t♦ ♣♦r s✉❛ ❛♣❛r❡♥t❡ s✐♠♣❧✐❝✐❞❛❞❡ ❝♦♠♦ ♣❡❧❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ ❢♦r♠❛s ❡♠ q✉❡ ♣♦❞❡ s❡r ❛♣r❡s❡♥t❛❞❛✳ ▼❛✐s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❋✐❜♦♥❛❝❝✐ ✈✐❞❡ ❬✶✶❪✳
❱❡♠♦s t❛♠❜é♠ ♥❛ ♠❛t❡♠át✐❝❛ ❞✐s❝r❡t❛ ✉♠ ❡①❡♠♣❧♦ ❢❛♠♦s♦ ❞❡ r❡❝♦rrê♥❝✐❛✱ ❛tr✐✲ ❜✉í❞♦ ❛ ❋❧❛✈✐✉s ❏♦s❡♣❤✉s✱ ✉♠ ❢❛♠♦s♦ ❤✐st♦r✐❛❞♦r ❞♦ ♣r✐♠❡✐r♦ sé❝✉❧♦✱ q✉❡ ❞✉r❛♥t❡ ❛ ❣✉❡rr❛ ❏✉❞❛✐❝❛✱ s❡ ❡♥❝♦♥tr❛✈❛ ❡♥tr❡ ✉♠ ❜❛♥❞♦ ❞❡ 41❥✉❞❡✉s r❡❜❡❧❞❡s ❡♥❝✉rr❛❧❛❞♦s
♣❡❧♦s r♦♠❛♥♦s ❡♠ ✉♠❛ ❝❛✈❡r♥❛✳ ❙❡♠ ❝❤❛♥❝❡ ❞❡ ❢✉❣❛ ♦ ❣r✉♣♦ ❞❡❝✐❞❡ ♣❡❧❛ ♠♦rt❡ ❛♦ ✐♥✈és ❞♦ ❛♣r✐s✐♦♥❛♠❡♥t♦✱ ♦s r❡❜❡❧❞❡s ❢♦r♠❛♠ ✉♠ ❝ír❝✉❧♦ ❡ ❝♦♠❡ç❛r✐❛♠ ❛ ♣❛rt✐r ❞❡ ❝❡rt♦ ♣♦♥t♦ ♣✉❧❛r ❞✉❛s ♣❡ss♦❛s ❡ ❛ ♠❛t❛r ❛ t❡r❝❡✐r❛ ♣❡ss♦❛ ♥✉♠❛ ❞✐r❡çã♦ ✜①❛✱ ❛ ❡❧✐♠✐♥❛çã♦ ♣r♦❝❡❞❡ ❡♠ t♦r♥♦ ❞♦ ❝ír❝✉❧♦ q✉❡ ✐rá s❡ t♦r♥❛♥❞♦ ♠❡♥♦r ❝♦♥❢♦r♠❡ ❛s ♣❡ss♦❛s ♠♦rt❛s sã♦ r❡♠♦✈✐❞❛s✱ ❛té ♥ã♦ r❡st❛r ❛❧❣✉é♠ ✈✐✈♦✳ ❈♦♥t❛ ❛ ❧❡♥❞❛ q✉❡ ❣r❛✲ ç❛s ❛♦ t❛❧❡♥t♦ ♠❛t❡♠át✐❝♦ ❞❡ ❏♦s❡♣❤✉s✱ ♦ ♠❡s♠♦ ❝♦♥s❡❣✉✐✉ ❡s❝❛♣❛r ❞❡st❛ t♦❧✐❝❡ ❞♦ s✉✐❝í❞✐♦ q✉❛♥❞♦ ❡♥❝♦♥tr♦✉ ♦ ❧♦❝❛❧ ♥♦ ❝ír❝✉❧♦ ✐♥✐❝✐❛❧ ❡ q✉❡♠ s❡r✐❛ ♦ ú❧t✐♠♦ ❛ ❡s❝❛♣❛r✳ ❋❧❛✈✐✉s ❏♦s❡♣❤✉s ❛tr✐❜✉✐ ❡♠ s✉❛ ❜✐♦❣r❛✜❛ q✉❡ ❞❡✈✐❞♦ à s♦rt❡ ♦✉ ❛ ♠ã♦ ❞❡ ❉❡✉s✱ ❡❧❡ ❡ ♦✉tr♦ ❤♦♠❡♠ r❡s♦❧✈❡r❛♠ s❡ ❡♥tr❡❣❛r ❛♦s r♦♠❛♥♦s✱ ❢❛t♦ q✉❡ ✐♥❝❧✉✐ ✉♠❛ ✈❛r✐❛çã♦ ♥♦
✶✳✷✳ ❋Ó❘▼❯▲❆❙ P❖❙■❈■❖◆❆■❙ ❊ ▼➱❚❖❉❖❙ ❘❊❈❯❘❙■❱❖❙
♣r♦❜❧❡♠❛✱ ✉♠❛ ✈❡③ q✉❡ ❤✐st♦r✐❛❞♦r❡s ❝♦♥s✐❞❡r❛♠ ♦ t❛❧ ❤♦♠❡♠ ❝♦♠♦ ✉♠ ❝ú♠♣❧✐❝❡ ❞❡ ❏♦s❡♣❤✉s✱ ❝♦♠♦ ❛ ♠♦rt❡ ❡r❛ ❛❞♠✐♥✐str❛❞❛ ♣❡❧❛ ♣ró①✐♠❛ ❡s❝♦❧❤❛ ♥❛ ✜❧❛✱ ❛ ❢♦r♠❛ ❞❡ ❏♦s❡♣❤✉s ❡✈✐t❛r ♦ ❛t♦ ❞❡ ❛ss❛ss✐♥❛r ✉♠ ❝♦❧❡❣❛ ✐♠♣❧✐❝❛ q✉❡ ❡❧❡ ❝♦❧♦❝♦✉ ❛❧❣✉é♠ ❞❡ ❝♦♠✉♠ ❛❝♦r❞♦ ♣❛r❛ s❡ r❡♥❞❡r ♥❛ ♣❡♥ú❧t✐♠❛ ♣♦s✐çã♦✳ ❱✐❞❡ r❡❢❡rê❝✐❛ ❜✐❜❧✐♦❣rá✜❝❛ ❬✶✷❪✳ ❊ss❡s ♣r♦❜❧❡♠❛s ❢❛s❝✐♥❛♥t❡s q✉❡ ❡stã♦ ♥❡st❡ ❜r❡✈❡ ❝♦♠❡♥tár✐♦ ❤✐stór✐❝♦✱ ❡ ♦✉tr♦s ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡s✱ s❡rã♦ ❛♥❛❧✐s❛❞♦s ❞❡t❛❧❤❛❞❛♠❡♥t❡ ♥♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ❞♦ ❡st✉❞♦ ❞❛s r❡❧❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛s ❧✐♥❡❛r❡s ❞❡st❛ ❞✐ss❡rt❛çã♦✳
✶✳✷ ❋ór♠✉❧❛s P♦s✐❝✐♦♥❛✐s ❡ ▼ét♦❞♦s ❘❡❝✉rs✐✈♦s
❯♠❛ s❡q✉ê♥❝✐❛ ✭✐♥✜♥✐t❛✮ ❞❡ ♥ú♠❡r♦s r❡❛✐s é ✉♠❛ ❧✐st❛ ❞❡ ✐♥✜♥✐t♦s ♥ú♠❡r♦s r❡✲ ❛✐s (a1, a2, a3, ...)✱ ♦✉ s❡❥❛✱ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♥❛ q✉❛❧ ✐❞❡♥t✐✜❝❛♠♦s q✉❡♠
é ♦ ♣r✐♠❡✐r♦ ♥ú♠❡r♦ ❞❛ s❡q✉ê♥❝✐❛✱ ♦ s❡❣✉♥❞♦✱ q✉❡♠ é ♦ t❡r❝❡✐r♦✱ ❡ ❛ss✐♠ s✉❝❡ss✐✲ ✈❛♠❡♥t❡✳ ❱❛♠♦s ❞❡♥♦t❛r ✉♠❛ s❡q✉ê♥❝✐❛ ✐♥✜♥✐t❛ ❝♦♠♦ ❛❝✐♠❛ ❝✐t❛❞❛ ♣♦r (ak)k>1 ♦✉
s✐♠♣❧❡s♠❡♥t❡ ♣♦r (ak)✳
❏á ✉♠❛ s❡q✉ê♥❝✐❛ ✭✜♥✐t❛✮ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ✐st♦ é✱ ✉♠❛ s❡q✉ê♥❝✐❛ ♦r❞❡♥❛❞❛ ✜♥✐t❛
(a1, a2, a3, ..., an) ❞❡ ♥ú♠❡r♦s✱ ❛♥❛❧♦❣❛♠❡♥t❡ ❛s s❡q✉ê♥❝✐❛s ✐♥✜♥✐t❛s✱ ✐❞❡♥t✐✜❝❛♠♦s q✉❡♠ é ♦ ♣r✐♠❡✐r♦ ♥ú♠❡r♦ ❞❛ s❡q✉ê♥❝✐❛✱ ♦ s❡❣✉♥❞♦✱ q✉❡♠ é ♦ t❡r❝❡✐r♦✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡ ❛té ♦ út✐♠♦ t❡r♠♦ an✳ ❚❛♠❜é♠ ♣♦❞❡♠♦s ❞❡♥♦t❛r ✉♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ ❝♦♠♦
✈✐♠♦s ✐♠❡❞✐❛t❛♠❡♥t❡ ♣♦r (ak)16k6n ♦✉ s✐♠♣❧❡s♠❡♥t❡ ♣♦r (ak)✱ ❡st❛ ú❧t✐♠❛ ❢♦r♠❛ s❡rá ♣♦r ❝♦♥✈❡♥✐ê♥❝✐❛ ❛ ♥♦t❛çã♦ ❛❞♦t❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦✱ ♦♥❞❡ ak é ♦ k♦ t❡r♠♦ ❞❛ s❡q✉ê♥❝✐❛✳
◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ♠♦str❛r ❛❧❣✉♥s t✐♣♦s ❡❧❡♠❡♥t❛r❡s ❞❡ s❡q✉ê♥❝✐❛s ❝♦♠ s✉❛s ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ✈á❧✐❞❛s ♣❛r❛ s❡q✉ê♥❝✐❛s ❡♠ ❣❡r❛❧✳ ◗✉❛♥❞♦ ❛ s❡q✉ê♥❝✐❛ ❢♦r ✜♥✐t❛ t♦♠❛r❡♠♦s ♦ ❝✉✐❞❛❞♦ ❞❡ ❞✐❢❡r❡♥❝✐á✲❧❛ ♣❛r❛ ♥ã♦ ❝♦♥❢✉♥❞✐ ❝♦♠ s❡q✉ê♥❝✐❛s ✐♥✜✲ ♥✐t❛s✳
❉✐③❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛(ak)❝♦♠k >1❡stá ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ ❢ór♠✉❧❛ ♣♦s✐❝✐♦♥❛❧ s❡ ♦s ✈❛❧♦r❡s ❞❡ ak ∈ R ❢♦r❡♠ ❞❛❞♦s ♣♦r ✉♠❛ ❢ór♠✉❧❛ q✉❡ ❞❡♣❡♥❞❡ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❛ ♣♦s✐çã♦ k✳ P❛r❛ ♠❡❧❤♦r ❡♥t❡♥❞❡r ✈❛♠♦s ♦❜s❡r✈❛r ♦ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r✳
❊①❡♠♣❧♦ ✶✳✶ ❆ s❡q✉ê♥❝✐❛(ak)❞♦s q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s é ❛ s❡q✉ê♥❝✐❛ (12,22,32, ...)✳ P♦rt❛♥t♦ t❡♠♦s a1 = 12✱ a2 = 22✱ a3 = 32 ❡✱ ♠❛✐s ❣❡r❛❧♠❡♥t❡✱ ak =k2 ♣❛r❛ k > 1 ✐♥t❡✐r♦✳
❖❜s❡r✈❛çã♦ ✶✳✶ ❊♠ ❛❧❣✉♥s ❝❛s♦s✱ ♣❛r❛ s✐♠♣❧✐✜❝❛r ❛ ❢ór♠✉❧❛ ♣♦s✐❝✐♦♥❛❧ q✉❡ ❞❡✜♥❡ ♦s ✈❛❧♦r❡s ❞♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛✱ é ✐♥t❡r❡ss❛♥t❡ ♣❡♥s❛r ♥❛s s❡q✉ê♥❝✐❛s ❞❡✜♥✐❞❛s ♣♦r ❢ór♠✉❧❛s ♣♦s✐❝✐♦♥❛✐s ❝♦♠❡ç❛♥❞♦ ♣❡❧♦ í♥❞✐❝❡ k = 0✳ P♦r ❡①❡♠♣❧♦✱ ❛ s❡q✉ê♥❝✐❛
❞❛s ♣♦tê♥❝✐❛s ✐♥t❡✐r❛s ♥ã♦ ♥❡❣❛t✐✈❛s ❞❡ 2✱ (1,2,4,8, . . .)✱ ♣♦❞❡ s❡r t❛♥t♦ r❡♣r❡s❡♥t❛❞❛
♣❡❧❛ s❡q✉ê♥❝✐❛ ❞❡ t❡r♠♦ ❣❡r❛❧ ak = 2k−1 ❝♦♠ k >1 q✉❛♥t♦ ♣❡❧❛ s❡q✉ê♥❝✐❛ ❞❡ t❡r♠♦ ❣❡r❛❧ bk = 2k ❝♦♠ k > 0✳ ◆♦t❡ q✉❡ ❛ ú❧t✐♠❛ ❢ór♠✉❧❛ é ❧✐❣❡✐r❛♠❡♥t❡ ♠❛✐s s✐♠♣❧❡s q✉❡ ❛ ❛♥t❡r✐♦r✳
✶✳✸✳ P❘❖●❘❊❙❙Õ❊❙ ❆❘■❚▼➱❚■❈❆❙
❖✉tr♦ ♣r♦❝❡❞✐♠❡♥t♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❞❡✜♥✐r ✉♠❛ s❡q✉ê♥❝✐❛ é ♦ ♠ét♦❞♦ r❡❝✉rs✐✈♦ q✉❡ ❝♦♥s✐st❡ ❡♠ ❞❡✜♥✐r ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ q✉❡ ❝❛❞❛ t❡r♠♦✱ ❛ ♣❛rt✐r ❞❡ ✉♠ ❝❡rt♦ í♥❞✐❝❡ k0✱ é ♦❜t✐❞♦ ❛tr❛✈és ❞♦s t❡r♠♦s ❛♥t❡r✐♦r❡s ❛ ❡❧❡✳ ❊st❡ ♠ét♦❞♦ ❢❛③ ♣❛rt❡ ❞♦
❡st✉❞♦ ❞❛s r❡❝♦rrê♥❝✐❛s q✉❡ s❡rá ❛❜♦r❞❛❞♦ ❝♦♠ ❞❡t❛❧❤❡s ♥♦ ❝❛♣ít✉❧♦ ✷✳ ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦✳
❊①❡♠♣❧♦ ✶✳✷ ❈♦♥s✐❞❡r❡ ❛ s❡q✉ê♥❝✐❛ (ak) ❝♦♠ k >1 ❞❡✜♥✐❞❛ ♣♦r a1 = 2✱ a2 = 5 ❡
ak= 2ak−1−ak−2, ∀k >3. ✭✶✳✶✮
❋❛③❡♥❞♦ k = 3 ♥❛ r❡❧❛çã♦ ❛❝✐♠❛✱ ♦❜t❡♠♦s
a3 = 2a2−a1 = 2.5−2 = 8;
❡ ❢❛③❡♥❞♦ k = 4✱ ♦❜t❡♠♦s
a4 = 2a3−a2 = 2.8−5 = 11,
❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ❆ r❡❧❛çã♦ ✭✶✳✶✮ é ✉♠❛ r❡❧❛çã♦ r❡❝✉rs✐✈❛ s❛t✐s❢❡✐t❛ ♣❡❧❛ s❡q✉ê♥❝✐❛ (ak) ❝♦♠ k > 1✳ ◆♦t❡ q✉❡ ❝❛❞❛ t❡r♠♦ é ❝❛❧❝✉❧❛❞♦ ❡♠ ❢✉♥çã♦ ❞♦s ❞♦✐s t❡r♠♦s ✐♠❡❞✐❛t❛♠❡♥t❡ ❛♥t❡r✐♦r❡s ❛ ❡❧❡✳ ❆ss✐♠✱ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ❞♦✐s ♣r✐♠❡✐r♦s t❡r♠♦s a1 = 2✱ a2 = 5 ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r t♦❞♦s ♦s ❞❡♠❛✐s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛✳
❯♠❛ ♣❡r❣✉♥t❛ ✐♥t❡r❡ss❛♥t❡ é ❛ s❡❣✉✐♥t❡✿ é ♣♦ssí✈❡❧ ♦❜t❡r ✉♠❛ ❢ór♠✉❧❛ ♣♦s✐❝✐♦♥❛❧ q✉❡ ❞❡✜♥❛ ❛ s❡q✉ê♥❝✐❛ ❛❝✐♠❛❄ ❆ r❡s♣♦st❛ é ♣♦s✐t✐✈❛ ❡ s❡rá ❞❛❞❛ ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✳ ❱❛❧❡ ❞❡st❛❝❛r t❛♠❜é♠ q✉❡ ♦ ♠ét♦❞♦ r❡❝✉rs✐✈♦ é ♠✉✐t♦ ✉t✐❧✐③❛❞♦ ♥❛s r❡s♦❧✉çõ❡s ❞❡ ♣r♦❜❧❡♠❛s ✉t✐❧✐③❛♥❞♦ ❛s ✐t❡r❛çõ❡s✳
❆❣♦r❛✱ ✈❛♠♦s ♣❛ss❛r ❛♦ ❡st✉❞♦ ❞❡ ❞♦✐s t✐♣♦s ❡s♣❡❝✐❛✐s ❞❡ s❡q✉ê♥❝✐❛s q✉❡ sã♦ ✈✐st❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❜r❛s✐❧❡✐r♦✱ ❛s ❝❤❛♠❛❞❛s ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ✭P❆✮ ❡ ♣r♦❣r❡ssõ❡s ❣❡♦♠étr✐❝❛s ✭P●✮✳
✶✳✸ Pr♦❣r❡ssõ❡s ❆r✐t♠ét✐❝❛s
❈♦♥t✐♥✉❛r❡♠♦s ♦ ❡st✉❞♦ ❞❡ s❡q✉ê♥❝✐❛s ❡❧❡♠❡♥t❛r❡s ❞❡✜♥✐♥❞♦ ❡ ❞✐s❝✉t✐♥❞♦ ❛s ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s✳
❉❡✜♥✐çã♦ ✶✳✶ ❯♠❛ s❡q✉ê♥❝✐❛ (ak) ❝♦♠ k > 1 ❞❡ ♥ú♠❡r♦s r❡❛✐s é ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ P❆ s❡ ❡①✐st✐r ✉♠ ♥ú♠❡r♦ r❡❛❧ r t❛❧ q✉❡ ❛ ❡q✉❛çã♦ r❡❝✉rs✐✈❛
ak+1 =ak+r ✭✶✳✷✮
s❡❥❛ s❛t✐s❢❡✐t❛ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ k >1✳
✶✳✸✳ P❘❖●❘❊❙❙Õ❊❙ ❆❘■❚▼➱❚■❈❆❙
❖ ♥ú♠❡r♦ r❡❛❧ r ❝❤❛♠❛✲s❡ r❛③ã♦ ❞❛ P❆✱ ❡❧❛ é ❛ ❞✐❢❡r❡♥ç❛ ❝♦♠✉♠ ❡♥tr❡ ❞♦✐s
t❡r♠♦s ❝♦♥s❡❝✉t✐✈♦s✱ ❛❧é♠ ❞✐ss♦ ♣❛r❛ q✉❡ ✉♠❛ P❆ s❡❥❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ é ♣r❡❝✐s♦ ❝♦♥❤❡❝❡r✱ ❛❧é♠ ❞❡ s✉❛ r❛③ã♦ r✱ t❛♠❜é♠ s❡✉ t❡r♠♦ ✐♥✐❝✐❛❧ a1✳
➱ ❝♦♠✉♠ ✈❡r ♥♦ ❞✐❛✲❞✐❛ s✐t✉❛çõ❡s ❡ ♣r♦❜❧❡♠❛s ❝♦♠ ❣r❛♥❞❡③❛s q✉❡ s♦❢r❡♠ ❛✉♠❡♥✲ t♦s ✐❣✉❛✐s ❡♠ ✐♥t❡r✈❛❧♦s ❞❡ t❡♠♣♦s ✐❣✉❛✐s✳ ❱❡❥❛♠♦s ❞♦✐s ❡①❡♠♣❧♦s q✉❡ r❡♣r❡s❡♥t❛♠ t❛✐s s✐t✉❛çõ❡s✳
❊①❡♠♣❧♦ ✶✳✸ ❯♠❛ ❞❡t❡r♠✐♥❛❞❛ ✐♥❞ústr✐❛ ❝♦♠❡ç♦✉ ❡♠ ❥❛♥❡✐r♦ ❞❡ 2015 s✉❛ ♣r♦❞✉✲
çã♦ ♠❡♥s❛❧ ❞❡ 10000t♦♥❡❧❛❞❛s ❡ ❢♦r♥❡❝❡✉ ❝♦♠ó❞✐t❡s ❛❣rí❝✉❧❛s ❛♦ ❣♦✈❡r♥♦ ❜r❛s✐❧❡✐r♦✳
❈♦♠ ❛ ❛❧t❛ ❞❡♠❛♥❞❛✱ ❡❧❛ ❛✉♠❡♥t♦✉ ❡♠ 10000 t♦♥❡❧❛❞❛s ♣♦r ♠ês s✉❛ ♣r♦❞✉çã♦ ❞✉✲
r❛♥t❡ t♦❞♦ ❛♥♦ ❞❡ 2015✳ ❊♥❝♦♥tr❡♠♦s ✉♠❛ ❢ór♠✉❧❛ r❡❝✉rs✐✈❛ q✉❡ ❞❡t❡r♠✐♥❡ ✉♠❛
P❆ t❡♥❞♦ ❝♦♠♦ t❡r♠♦ ✐♥✐❝✐❛❧ a1 ❡ r❛③ã♦ r✳
❙♦❧✉çã♦✿ ❆♥❛❧✐s❛♥❞♦ ❛s ✐♥❢♦r♠❛çõ❡s ❛❝✐♠❛✱ t❡♠♦s q✉❡ s❡ ❡♠ ❥❛♥❡✐r♦ ❞❡ 2015 ❛
♣r♦❞✉çã♦ ❞❡ ❝♦♠ó❞✐t❡s ❞❡st❛ ✐♥❞ústr✐❛ ❢♦✐ ❞❡ 10000 t♦♥❡❧❛❞❛s✱ ❧♦❣♦ ❡♠ s❡❣✉✐❞❛✱
♦❜❡❞❡❝❡♥❞♦ ❛ r❡❣r❛ ❡st❛❜❡❧❡❝✐❞❛✱ ❡♠ ❢❡✈❡r❡✐r♦ ❡ss❛ ♣r♦❞✉çã♦ ❢♦✐ ❞❡ 20000 t♦♥❡❧❛❞❛s✱
❡♠ ♠❛rç♦ s❡❣✉✐♥❞♦ ❛ ♠❡s♠❛ ♦r❞❡♠✱ ❢♦✐ ❞❡ 30000 t♦♥❡❧❛❞❛s✳
P♦rt❛♥t♦ ❛ s❡q✉ê♥❝✐❛ (ak) ❝♦♠ k >1✱ ♦♥❞❡ a1 = 10000 ❝✉❥❛ ❛ ❢ór♠✉❧❛ r❡❝✉rs✐✈❛
é ❞❛❞❛ ♣♦r ak+1 = ak+ 10000 ♣❛r❛ k > 1 é ✉♠❛ P❆ ❞❡ t❡r♠♦ ✐♥✐❝✐❛❧ a1 = 10000 ❡
r❛③ã♦ r= 10000✳ ⋄
❊①❡♠♣❧♦ ✶✳✹ ❈❛r❧♦s ❡st❛❜❡❧❡❝❡✉ ✉♠❛ ♠❡t❛ ♣❛r❛ ♦ ❛♥♦ ❞❡ 2016 ❡ ❝♦♠♦ ❤❛✈✐❛ ❡♠ 2015 ❡❝♦♥♦♠✐③❛❞♦ 1000 r❡❛✐s✱ ♣r❡❝✐s❛✈❛ ❝♦♥t✐♥✉❛r ♣♦✉♣❛♥❞♦✱ ♣♦✐s ❛ ❝r✐s❡ ❡❝♦♥ô♠✐❝❛
♥♦ ❇r❛s✐❧ ❡st❛✈❛ ❧❤❡ ❡♥s✐♥❛♥❞♦ ❛ ♣♦✉♣❛r✳ ➚ ♣❛rt✐r ❞♦ ❛❝✉♠✉❧❛❞♦ ❡♠ 2015 ❛s ❡❝♦✲
♥♦♠✐❛s ❞❡ ❈❛r❧♦s q✉❡ ❡♠ t❡s❡ ❝r❡s❝❡rá t♦❞♦ ♠ês 200 r❡❛✐s✱ ♠♦str❛rá ✉♠❛ ❢ór♠✉❧❛
r❡❝✉rs✐✈❛ q✉❡ ❞❡t❡r♠✐♥❛rá ✉♠❛ P❆ t❡♥❞♦ ❝♦♠♦ t❡r♠♦ ✐♥✐❝✐❛❧ a1 ❡ ❛ r❛③ã♦ r✳
❙♦❧✉çã♦✿ ❱❡♠♦s q✉❡ s❡ ❡♠ 2015 ❈❛r❧♦s ❡❝♦♥♦♠✐③♦✉ 1000 r❡❛✐s✱ ❡♠ ❥❛♥❡✐r♦ ❞❡ 2016
✜❝❛rá ❝♦♠ 1200✱ ❡♠ ❢❡✈❡r❡✐r♦✱ ❝♦♥❢♦r♠❡ ❛ ♠❡s♠❛ ♦r❞❡♠✱ t❡rá ❡❝♦♥♦♠✐③❛❞♦ 1400
r❡❛✐s ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ▲♦❣♦ t❡♠♦s ❛ s❡q✉ê♥❝✐❛ (ak)❝♦♠ k >1✱ ♦♥❞❡a1 = 1000 ❡
❛ ❢ór♠✉❧❛ r❡❝✉rs✐✈❛ é ❞❛❞❛ ♣♦r ak+1 =ak+ 200 ♣❛r❛k >1✳ ❊st❛ ❢ór♠✉❧❛ ❞❡t❡r♠✐♥❛ ✉♠❛ P❆ ❝✉❥♦ t❡r♠♦ ✐♥✐❝✐❛❧ é a1 = 1000 ❡ r❛③ã♦ r = 200✳ ⋄
◆♦t❡ q✉❡ s❡ ♥♦s ❞♦✐s ❡①❡♠♣❧♦s ú❧t✐♠♦s ❡①❡♠♣❧♦s t✐✈éss❡♠♦s ❛♣❡♥❛s ak+1 =ak+
10000♣❛r❛k >1❡ak+1 =ak+ 200♣❛r❛ k>1✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ss❛s ❡q✉❛çõ❡s ♥ã♦
❝❛r❛❝t❡r✐③❛r✐❛♠ ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s✱ ♣♦✐s ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧ ✐❞❡♥t✐✜❝❛r ♦ ✈❛❧♦r ❞♦ ♣r✐♠❡✐r♦ t❡r♠♦ ❡♠ ❝❛❞❛ ✉♠❛ ❞❡❧❛s✱ ✐st♦ é✱ ❝♦♥❤❡❝❡r ♦ t❡r♠♦ ✐♥✐❝✐❛❧ a1✳
❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r tr❛t❛ ❞❡ ♦✉tr❛ ❝❛r❛❝t❡r✐③❛çã♦ r❡❝✉rs✐✈❛ ♠✉✐t♦ út✐❧ ♣❛r❛ P❆✬s✳
Pr♦♣♦s✐çã♦ ✶✳✶ ❯♠❛ s❡q✉ê♥❝✐❛ (ak) ❝♦♠ k >1 ❞❡ ♥ú♠❡r♦s r❡❛✐s é ✉♠❛ P❆✱ s❡ ❡ só s❡✱
ak+2+ak= 2ak+1,∀k>1. ✭✶✳✸✮
✶✳✸✳ P❘❖●❘❊❙❙Õ❊❙ ❆❘■❚▼➱❚■❈❆❙
❉❡♠♦♥str❛çã♦✿ P♦r ❞❡✜♥✐çã♦✱ ❛ s❡q✉ê♥❝✐❛ é ✉♠❛ P❆✱ s❡ ❡ só s❡✱a2−a1 =a3−a2 =
...✱ ✐st♦ é✱ s❡ ❡ só s❡✱ ♣❛r❛ t♦❞♦ k >1✐♥t❡✐r♦✱ t✐✈❡r♠♦s ak+2−ak+1 =ak+1−ak✱ q✉❡
é ✉♠❛ ♠❛♥❡✐r❛ ❡q✉✐✈❛❧❡♥t❡ ❞❡ ❡s❝r❡✈❡r♠♦s ✭✶✳✸✮✳
❆❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ ✈❡r❡♠♦s ❛ s❡❣✉✐r tr❛t❛♠ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✉♠❛ P❆✱ ❡❧❡s ❡♥✉♥❝✐❛♠ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❝♦♠♦ ♦❜t❡r ✉♠❛ ❢ór♠✉❧❛ ♣♦s✐❝✐♦♥❛❧ ♣❛r❛ ♦s t❡r♠♦s ❞❡st❛ P❆✳
Pr♦♣♦s✐çã♦ ✶✳✷ ❙❡ (ak) ❝♦♠ k >1 é ✉♠❛ P❆ ❞❡ r❛③ã♦ r✱ ❡♥tã♦ ✭❛✮ ak =a1+ (k−1)r✱ ♣❛r❛ t♦❞♦ k >1✳
✭❜✮ a1+a2+...+ak =
k(a1+ak)
2 ✱ ♣❛r❛ t♦❞♦ k >1
❉❡♠♦♥str❛çã♦✿ ✭❛✮ ❖ ❞✐❛❣r❛♠❛
a1 +r / /a 2 +r / /a 3 +r /
/· · · +r //a
k−1
+r
/
/a
k
♠♦str❛ q✉❡ ♣❛r❛ ❝❤❡❣❛r ❛ ak ❛ ♣❛rt✐r ❞❡ a1✱ sã♦ ♥❡❝❡ssár✐♦sk−1♣❛ss♦s✱ ♦♥❞❡ ❝❛❞❛
❛✈❛♥ç♦ ❡q✉✐✈❛❧❡ ❛ s♦♠❛r r ❛ ✉♠ t❡r♠♦✳ P♦rt❛♥t♦✱ ♣❛r❛ ♦❜t❡r ak ❞❡✈❡♠♦s s♦♠❛r✱ ❛♦ t♦❞♦✱ (k−1)r ❛a1✱ ❞❡ ❢♦r♠❛ q✉❡ ak=a1+ (k−1)r✳
✭❜✮ ❆ ♣❛rt✐r ❞♦ ❞✐❛❣r❛♠❛
a1 +
r / /a 2 + r / /a 3 + r /
/· · · a
k−2 −r
o
o a
k−1 −r o o a k −r o o
t❡♠♦s ❞❛í q✉❡
a1+ak= (a2−r) + (ak−1+r) =a2+ak−1,
a2+ak−1 = (a3−r) + (ak−2+r) =a3+ak−2,
a3+ak−2 = (a4−r) + (ak−3+r) =a4+ak−3,
. . . .
P♦rt❛♥t♦✱ s❡♥❞♦ S =a1+a2+...+ak✱ t❡♠♦s
2S = 2(a1+a2+a3+...+ak−2+ak−1+ak)
= (a1+ak) + (a2+ak−1) + (a3+ak−2) +. . .+ (ak+a1)
= (a1+ak) + (a1+ak) + (a1+ak) +. . .+ (a1+ak)
| {z }
k ♣❛r❝❡❧❛s
= k(a1+ak),
✶✳✸✳ P❘❖●❘❊❙❙Õ❊❙ ❆❘■❚▼➱❚■❈❆❙
❡ ❝♦♥❝❧✉í♠♦s q✉❡
S = k(a1+ak)
2 ,
❞♦♥❞❡ t❡♠♦s ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❡♠ ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛✳
❆s ❢ór♠✉❧❛s ❞♦s ✐t❡♥s ✭❛✮ ❡ ✭❜✮ ❞❛ ♣r♦♣♦s✐çã♦ ✶✳✷ sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❢ór♠✉❧❛ ❞♦ t❡r♠♦ ❣❡r❛❧ ❡ ❛ ❢ór♠✉❧❛ ♣❛r❛ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ✉♠❛
P❆✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❊①❡♠♣❧♦ ✶✳✺ ❈❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s k ♣r✐♠❡✐r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s í♠♣❛r❡s✳
❙♦❧✉çã♦✿ ❖❜s❡r✈❡ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s í♠♣❛r❡s é 1,3,5,7, ...✱ é
✉♠❛ P❆ ❞❡ r❛③ã♦ 2✳ ▲♦❣♦ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✶✳✷✭❛✮✱ ♦ k✲és✐♠♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ é
ak = 1 + (k−1).2 = 2k−1.
❏á ❛ s♦♠❛ ❞♦s k ♣r✐♠❡✐r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s í♠♣❛r❡s é ♦❜t✐❞❛ ❛tr❛✈és ❞❛ ♣r♦♣♦✲
s✐çã♦ ✶✳✷✭❜✮✱ ♦✉ s❡❥❛
Sk=
k[1 + (2k−1)]
2 =k
2.
⋄
❊①❡♠♣❧♦ ✶✳✻ ❈♦♥s✐❞❡r❡ ❛ s❡q✉ê♥❝✐❛ (ak) ❝♦♠ k >1 ❞❛❞❛ ♣♦r a1 = 1 ❡ s❡❥❛
ak+1 =
ak
1 + 2ak
♣❛r❛ t♦❞♦ k >1 ✐♥t❡✐r♦✳ ❈❛❧❝✉❧❛r ak ❡♠ ❢✉♥çã♦ ❞❡ k✳
❙♦❧✉çã♦✿ ❈♦♠♦ t♦❞♦s ♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ sã♦ ♣♦s✐t✐✈♦s✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ s❡q✉ê♥❝✐❛ (bk)❝♦♠ k >1✱ ❢❛③❡♥❞♦bk= a1k✳ ❆ss✐♠ ✈❡r✐✜❝❛♠♦s q✉❡
bk+1 = 1
ak+1
= 1 + 2ak
ak
= 1
ak
+ 2 =bk+ 2,
❧♦❣♦✱ (bk) ❝♦♠ k>1 é ✉♠❛ P❆ ❝♦♠ t❡r♠♦ ✐♥✐❝✐❛❧ b1 = a11 = 1 ❡ r❛③ã♦ 2✳
◆♦ ❡♥t❛♥t♦ ❡st❛ P❆ é s✐♠✐❧❛r ❛ P❆ ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ❡♠ q✉❡ ♦s t❡r♠♦s sã♦ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s í♠♣❛r❡s✱ ♦♥❞❡ ✈✐♠♦s q✉❡ bk= 2k−1 ♣❛r❛ t♦❞♦ k >1✳
❉❛í t❡♠♦s q✉❡✱
ak=
1
bk
= 1
2k−1.
⋄
✶✳✹✳ P❘❖●❘❊❙❙Õ❊❙ ●❊❖▼➱❚❘■❈❆❙
✶✳✹ Pr♦❣r❡ssõ❡s ●❡♦♠étr✐❝❛s
❱❡r❡♠♦s ♥❡st❛ s❡çã♦ ♦✉tr❛ ❝❧❛ss❡ ❜❛st❛♥t❡ út✐❧ ❞❡ s❡q✉ê♥❝✐❛s ❢♦r♠❛❞❛ ♣❡❧❛s ♣r♦✲ ❣r❡ssõ❡s ❣❡♦♠étr✐❝❛s✳
❉❡✜♥✐çã♦ ✶✳✷ ❯♠❛ s❡q✉ê♥❝✐❛ (ak) ❝♦♠ k > 1 ❞❡ ♥ú♠❡r♦s r❡❛✐s é ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ✭P●✮ s❡ ❡①✐st✐r ✉♠ ♥ú♠❡r♦ r❡❛❧ q t❛❧ q✉❡ ❛ ❢ór♠✉❧❛ r❡❝✉rs✐✈❛
ak+1 =q.ak ✭✶✳✹✮
s❡❥❛ s❛t✐s❢❡✐t❛ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ k >1✳
❙❡♠❡❧❤❛♥t❡ às P❆✬s✱ ♦ ♥ú♠❡r♦ r❡❛❧ q q✉❡ ❛♣❛r❡❝❡ ♥❛ ❞❡✜♥✐çã♦ ❞❡ P● é ❛ r❛③ã♦
❞❛ ♠❡s♠❛✳ ❖❜s❡r✈❡♠♦s q✉❡✿
❙❡ q= 0✱ ❡♥tã♦ak = 0♣❛r❛ t♦❞♦k ≥1✳ ❙❡ ♣♦r ♦✉tr♦ ❧❛❞♦q = 1✱ ❡♥tã♦ak+1 =ak ♣❛r❛ t♦❞♦ k >1✳
❆ss✐♠✱ ✉♠❛ P● (ak) ❝♦♠ k > 1 só s❡rá ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ s❡ ❞❡❧❛ ❝♦♥❤❡❝❡r♠♦s ♦ ♣r✐♠❡✐r♦ t❡r♠♦ a1 ❡ ❛ r❛③ã♦ q✳
P❛r❛ ♥♦ss❛ ❝♦♠♣r❡❡♥sã♦ s♦❜r❡ P● ✈❡r❡♠♦s ❛❞✐❛♥t❡ ❛❧❣✉♠❛s ♣r♦♣♦s✐çõ❡s ❡ s✉❛s ❞❡♠♦♥str❛çõ❡s✳
Pr♦♣♦s✐çã♦ ✶✳✸ ❙❡❥❛♠ q ❡ a r❡❛✐s ♣♦s✐t✐✈♦s ❡ n ♥❛t✉r❛❧✳ ❚❡♠♦s q✉❡
✭❛✮ ❙❡ 0< q <1✱ ❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ (an) ❞❡ t❡r♠♦ ❣❡r❛❧ an =aqn é ❞❡❝r❡s❝❡♥t❡✳
✭❜✮ ❙❡ q❃✶✱ ❡♥tã♦ ❛ ♠❡s♠❛ s❡q✉ê♥❝✐❛ (an) é ❝r❡s❝❡♥t❡✳ ❉❡♠♦♥str❛çã♦✿
✭❛✮ ❈♦♠♦q ❡a sã♦ ♣♦s✐t✐✈♦s✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r aq ❛♠❜♦s ♦s ♠❡♠❜r♦s s❡♥❞♦ q <1✱
♦❜t❡♠♦s
aq2 < aq,
♠✉❧t✐♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♣♦r q✱ s❡❣✉❡
aq3 < aq2,
❡ ❞❛í
aq3 < aq2 < aq.
❈♦♥t✐♥✉❛♥❞♦✱ ❝❤❡❣❛♠♦s ❛♦ r❡s✉❧t❛❞♦ ❛❧♠❡❥❛❞♦✱ ♦✉ s❡❥❛❀
· · ·< aq4 < aq3 < aq2 < aq,
❧♦❣♦✱ ❛ s❡q✉ê♥❝✐❛ (an)❞❡ t❡r♠♦ ❣❡r❛❧ an=aqn é ❞❡❝r❡s❝❡♥t❡✳
✶✳✹✳ P❘❖●❘❊❙❙Õ❊❙ ●❊❖▼➱❚❘■❈❆❙
❊♠ s❡❣✉✐❞❛ t❡♠♦s q✉❡ ✭❜✮✱ ✐❞❡♥t✐❝❛♠❡♥t❡ ❛♦ ✐t❡♠ ✭❛✮✱ ❝♦♠♦ a ❡q sã♦ ♣♦s✐t✐✈♦s✱
♠✉❧t✐♣❧✐❝❛♠♦s ♣♦r aq ❛♠❜♦s ♦s ♠❡♠❜r♦s✱ q >1✱ ♦❜t❡♠♦s
aq2 > aq,
♠✉❧t✐♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♣♦r q✱ s❡❣✉❡
aq3 > aq2,
❡ ❞❛í
aq3 > aq2 > aq.
❈♦♥t✐♥✉❛♥❞♦✱ ❝❤❡❣❛♠♦s ❛♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✱ ♦✉ s❡❥❛❀
· · ·> aq4 > aq3 > aq2 > aq,
❧♦❣♦✱ ❛ s❡q✉ê♥❝✐❛ (an)❞❡ t❡r♠♦ ❣❡r❛❧ an=aqn é ❝r❡s❝❡♥t❡✳
❊♠ s❡❣✉✐❞❛✱ ✈❡r❡♠♦s ♥❛ ♣r♦♣♦s✐çã♦ ♦✉tr❛ ❝❛r❛❝t❡r✐③❛çã♦ r❡❝✉rs✐✈❛ út✐❧ ♣❛r❛ ❛ ♠❛✐♦r✐❛ ❞❛s P●✬s✳
Pr♦♣♦s✐çã♦ ✶✳✹ ❯♠❛ s❡q✉ê♥❝✐❛ (ak) ❝♦♠ k >1❞❡ ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ♥✉❧♦s é ✉♠❛ P● s❡ ❡ só s❡
ak+2ak =ak2+1,∀k>1. ✭✶✳✺✮
❉❡♠♦♥str❛çã♦✿ ❆ s❡q✉ê♥❝✐❛ é ✉♠❛ P✳● ❞❡ r❛③ã♦ q✱ ♣♦r ❞❡✜♥✐çã♦✱ s❡ ❡ só s❡ a2
a1
= a3
a2
= a4
a3
=· · ·=q,
♦✉ s❡❥❛✱ s❡ ❡ só s❡✱ ♣❛r❛ t♦❞♦ k>1 ✐♥t❡✐r♦✱ t✐✈❡r♠♦s
ak+2
ak+1
= ak+1
ak q✉❡ é ✉♠❛ ♠❛♥❡✐r❛ ♣❛r❡❝✐❞❛ ❞❡ ❡s❝r❡✈❡r ✭✶✳✺✮✳
❆❣♦r❛✱ s❡❣✉✐♥❞♦ ❛ ♠❡s♠❛ ❧ó❣✐❝❛ ❞♦ ❡st✉❞♦ ❞❛s P❆✬s✱ ♦ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐r tr❛rá ❛s ❢ór♠✉❧❛s ♣❛r❛ ♦ t❡r♠♦ ❣❡r❛❧ ❡ ♣❛r❛ ❛ s♦♠❛ ❞❡ k ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ✉♠❛ P●✳
Pr♦♣♦s✐çã♦ ✶✳✺ ❙❡ (ak) ❝♦♠ k >1 é ✉♠❛ P● ❞❡ r❛③ã♦ q✱ ❡♥tã♦✿ ✭❛✮ ak =a1.qk−1✱ ♣❛r❛ t♦❞♦ k>1✳
✭❜✮ ❙❡ q 6=✶✱ ❡♥tã♦ a1+a2+...+ak =a1(q
k
−1)
q−1 ✱ ♣❛r❛ t♦❞♦ k >1.
✶✳✹✳ P❘❖●❘❊❙❙Õ❊❙ ●❊❖▼➱❚❘■❈❆❙
❉❡♠♦♥str❛çã♦✿ ✭❛✮ ◆♦ ❞✐❛❣r❛♠❛
a1 .q / /a 2 .q / /a 3 .q / /· · · .q / /a
k−1
.q
/
/a
k
♣❛r❛ ❝❤❡❣❛r ❛ ak à ♣❛rt✐r ❞❡ a1✱ é ♣r❡❝✐s♦ k−1 ♣❛ss♦s✱ ♦♥❞❡ ❝❛❞❛ ♣❛ss♦ s❡ r❡s✉♠❡
❛ ♠✉❧t✐♣❧✐❝❛r ✉♠ t❡r♠♦ ♣♦r q✳ P♦rt❛♥t♦✱ t❡♠♦s ❞❡ ♠✉❧t✐♣❧✐❝❛r a1 ♣♦r q ✉♠ t♦t❛❧ ❞❡
k−1✈❡③❡s✱ ❡ ❞❛í
ak=a1.qk−1.
✭❜✮ ❉❡♥♦t❛♠♦s ♣♦r Sk ❛ s♦♠❛ ❞❡s❡❥❛❞❛✱ t❡♠♦s q✉❡✱ Sk = a1+a2+...+ak✱ s❡❣✉❡
❡♥tã♦ ❞❡ ✭✶✳✹✮ q✉❡
qSk = q(a1+a2+a3+...+ak−2+ak−1+ak)
= qa1+qa2+qa3+. . .+qak−1+qak−2+qak
= a2+a3+...+ak−1+ak+ak+1.
❧♦❣♦✱ s✉❜tr❛✐♥❞♦ ❛ ❡①♣r❡ssã♦ ❞❡s❡♥✈♦❧✈✐❞❛ ❛❝✐♠❛ ♣♦r (Sk) t❡♠♦s✱
(q−1)Sk = qSk−Sk
= (a2+a3 +...+ak+ak+1)−(a1+a2+...+ak)
= (a2+a3 +...+ak) +ak+1−a1−(a2+...+ak)
= (a2+a3 +...+ak)−(a2+...+ak) +ak+1−a1
= ak+1−a1.
❆❣♦r❛ ❞✐✈✐❞✐♠♦s ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ✐❣✉❛❧❞❛❞❡ (q−1)Sk =ak+1−a1 ♣♦r q−1✳
❉❛í t❡♠♦s
Sk=
ak+1−a1
q−1 =
a1qk−a1
q−1 =
a1(qk−1)
q−1 ,
❝♦♠♦ q✉❡r❡♠♦s ❞❡♠♦♥str❛r✳
❊①❡♠♣❧♦ ✶✳✼ ❉✐③ ❛ ❧❡♥❞❛ q✉❡ ♦ ✐♥✈❡♥t♦r ❞♦ ①❛❞r❡③ ♣❡❞✐✉ ❝♦♠♦ r❡❝♦♠♣❡♥s❛ 1 ❣rã♦
❞❡ tr✐❣♦ ♣❡❧❛ ♣r✐♠❡✐r❛ ❝❛s❛✱ 2❣rã♦s ♣❡❧❛ s❡❣✉♥❞❛✱ 4 ♣❡❧❛ t❡r❝❡✐r❛ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱
s❡♠♣r❡ ❞♦❜r❛♥❞♦ ❛ q✉❛♥t✐❞❛❞❡ ❛ ❝❛❞❛ ❝❛s❛ ♥♦✈❛✳ ❈❛❧❝✉❧❡♠♦s ❡♥tã♦ ❛ r❡❝♦♠♣❡♥s❛ ❞♦ ✐♥✈❡♥t♦r ❞♦ ①❛❞r❡③✿
❙♦❧✉çã♦✿ ❈♦♠♦ ♦ t❛❜✉❧❡✐r♦ ❞❡ ①❛❞r❡③ t❡♠ 64 ❝❛s❛s✱ ♦ ♥ú♠❡r♦ ❞❡ ❣rã♦s ♣❡❞✐❞♦s
♣❡❧♦ ✐♥✈❡♥t♦r ❞♦ ❥♦❣♦ é ❛ s♦♠❛ ❞♦s 64 ♣r✐♠❡✐r♦s t❡r♠♦s ❞❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ 1,2,4, ...✳ ❖ ✈❛❧♦r ❞❡ss❛ s♦♠❛ é
Sk =a1
1−qk
1−q = 1
1−264
1−2 = 2
64
−1.
✶✳✹✳ P❘❖●❘❊❙❙Õ❊❙ ●❊❖▼➱❚❘■❈❆❙
⋄ ❈❛❧❝✉❧❛♥❞♦✱ ♦❜t❡♠♦s 18446744073709551615✳
❊①❡♠♣❧♦ ✶✳✽ ❈♦♥s✐❞❡r❡ ❛ s♦♠❛ ❞❛ P● ✐♥✜♥✐t❛ 0,3 + 0,03 + 0,003 +... ❡♠ q✉❡
|q|<1✳
❙♦❧✉çã♦✿ ❈♦♠ ❜❛s❡ ♥❡st❡ ❡①❡♠♣❧♦ ✈❛♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❛ s♦♠❛ ❞❡ ✉♠❛ P● ✐♥✜♥✐t❛✳ ❖❜s❡r✈❡ q✉❡ a1 = 0,3 ❡ q= 0,1✱ ❧♦❣♦ s✉❜st✐t✉✐♥❞♦ ♥❛ ❢ór♠✉❧❛ ❞❛
s♦♠❛ ❞♦s t❡r♠♦s ❞❡ ✉♠❛ P● ✜♥✐t❛✱ t❡♠♦s
Sk =
ak.q−a1
q−1 =
ak.0,1−0,3
0,1−1 =
3−ak
9 .
❉❛í s❡❣✉❡✱ q✉❛♥❞♦ k= 2✱ ❡♥tã♦
S2 =
3−a2
9 =
3−0,03
9 = 0,33.
◗✉❛♥❞♦ k = 3✱ ❡♥tã♦
S3 =
3−a3
9 =
3−0,003
9 = 0,333.
◗✉❛♥❞♦ k = 4✱ ❡♥tã♦
S4 =
3−a4
9 =
3−0,0003
9 = 0,3333.
◗✉❛♥❞♦ k = 5✱ ❡♥tã♦
S5 =
3−a5
9 =
3−0,00003
9 = 0,33333.
P♦❞❡♠♦s ✈❡r q✉❡ q✉❛♥t♦ ♠❛✐♦r ❢♦r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ t❡r♠♦s ❞❛ P●✱ ♠❛✐s ♣ró①✐♠♦ ❞❡ ③❡r♦ s❡ t♦r♥❛ ak✱ ✐st♦ é3−ak ✜❝❛ ♠❛✐s ♣ró①✐♠♦ ❞❡ 3✱ ❝♦♥❝❧✉í♠♦♦s✱ ♣♦rt❛♥t♦ q✉❡ ❛ s♦♠❛ ❞♦s ✐♥✜♥✐t♦s t❡r♠♦s ❞❡ss❛ P● é ❛ ❞í③✐♠❛ ♣❡r✐ó❞✐❝❛ 0,33333...= 13. ❉❡ ♠♦❞♦
❣❡r❛❧✱ ❛ s♦♠❛ ❞♦s t❡r♠♦s ❞❡ ✉♠❛ P● ✐♥✜♥✐t❛ é ❞❛❞❛ ♣♦r
S= lim
k→∞Sk =
a1
1−q.
✐st♦ é✱
S = a1
1−q.
⋄ ❆✐♥❞❛ ♥♦s r❡❢❡r✐♥❞♦ ❛ P● ❝♦♠♦ s♦♠❛ ✐♥✜♥✐t❛✱ t❡♠♦s ♥♦ ❡①❡♠♣❧♦ ❛❜❛✐①♦ ❛ s❡❣✉✐♥t❡ s✐t✉❛çã♦✲♣r♦❜❧❡♠❛✿
✶✳✹✳ P❘❖●❘❊❙❙Õ❊❙ ●❊❖▼➱❚❘■❈❆❙
❊①❡♠♣❧♦ ✶✳✾ ❙✉♣♦♥❤❛ q✉❡ ✉♠ ❛t❧❡t❛ ❡♠ ✉♠ tr❡✐♥♦ ❞❡ ❈♦rr✐❞❛ ❞❡✈❛ ❝♦rr❡r 1 ❦♠✳
■♥✐❝✐❛❧♠❡♥t❡ ❡❧❡ ❝♦rr❡ ♠❡t❛❞❡ ❞❡ss❛ ❞✐stâ♥❝✐❛✱ ✐st♦ é✱ 1
2 ❦♠❀ ❡♠ s❡❣✉✐❞❛ ❡❧❡ ❝♦rr❡
♠❡t❛❞❡ ❞❛ ❞✐stâ♥❝✐❛ q✉❡ ❢❛❧t❛✱ ✐st♦ é✱ 1
4 ❦♠❀ ❞❡♣♦✐s ♠❡t❛❞❡ ❞❛ ❞✐stâ♥❝✐❛ r❡st❛♥t❡✱
✐st♦ é✱ 1
8 ❦♠✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❉❡♣♦✐s ❞❡ n ❡t❛♣❛s✱ q✉❛♥t♦s ♠❡tr♦s t❡rá ❝♦rr✐❞♦
❡ss❡ ❛t❧❡t❛❄
❙♦❧✉çã♦✿ ❘❡s♦❧✈❡♥❞♦ ❡ss❡ ♣r♦❜❧❡♠❛✱ ✈❡♠♦s q✉❡ ♦ ❛t❧❡t❛ t❡rá ♣❡r❝♦rr✐❞♦✱
1
2+
1
4+
1
8+...+
1
2k km.
❙❡ k ❢♦r ❣r❛♥❞❡✱ ❡ss❛ s♦♠❛ s❡rá ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐❣✉❛❧ ❛ 1❦♠ ❡ ❞❛í ❝❛❧❝✉❧❛♠♦s
❛ s♦♠❛ ❞❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛
S = 1
2 + 1 4 + 1 8 + 1
16 +....
❚❡♠♦s ❡♥tã♦ q✉❡
S = a1
1−q =
1 2
1−12 = 1.
❖ q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ ♦ ✈❛❧♦r ❥á ❡s♣❡r❛❞♦ ❞❡ 1 ❦♠✳ ⋄
❊①❡♠♣❧♦ ✶✳✶✵ ❙❡❥❛ (ak) ❝♦♠ k >1 ✉♠❛ P❆ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❞❡ r❛③ã♦ r > 0✱ ❡ (bk) ❝♦♠ k > 1 ✉♠❛ P● ❞❡ ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ♥✉❧♦s ❞❡ r❛③ã♦ q✳ ❈♦♥s✐❞❡r❡ ❛ s❡q✉ê♥❝✐❛ (ck) ❝♦♠ k > 1 t❛❧ q✉❡ ck =bak ♣❛r❛ t♦❞♦ k >1 ✐♥t❡✐r♦✳ Pr♦✈❛r q✉❡ (ck)
❝♦♠ k>1 é ✉♠❛ P✳● ❞❡ r❛③ã♦ qr✳
❙♦❧✉çã♦✿ P❛r❛ ♣r♦✈❛r q✉❡ ❛ r❛③ã♦ ❡♥tr❡ ❞♦✐s t❡r♠♦s ❝♦♥s❡❝✉t✐✈♦s ❞❛ s❡q✉ê♥❝✐❛ (ck) ❝♦♠ k >1é s❡♠♣r❡ ✐❣✉❛❧ ❛ qr✱ ❜❛st❛ ♠♦str❛r q✉❡
ck+1
ck
= bak+1
bak
= b1q
ak+1−1
b1qak−1
=qak+1−ak
=qr.
⋄
❊①❡♠♣❧♦ ✶✳✶✶ ❈❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❛ s♦♠❛
2.1 + 7.3 + 12.32+ 17.33+...+ 497.399+ 502.3100,
♦♥❞❡✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❛ k❛ ♣❛r❝❡❧❛ é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞♦ k♦ t❡r♠♦ ❞❛ P❆
(2,7,12, ...,502) ♣❡❧♦ k♦ t❡r♠♦ ❞❛ P● (1,3,32, ...,3100)✳
✶✳✹✳ P❘❖●❘❊❙❙Õ❊❙ ●❊❖▼➱❚❘■❈❆❙
❙♦❧✉çã♦✿ ❙❡❣✉✐♥❞♦ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ❞❛ ❞❡♠♦♥str❛çã♦✱ ❞❡♥♦t❛♠♦s ♣♦r Sk ❛ s♦♠❛ ♣❡❞✐❞❛ ❡ ❝♦♠♦ 3 é ❛ r❛③ã♦ ❞❛ P●✱ ❝❛❧❝✉❧❛♠♦s ❞❛í ♦ ✈❛❧♦r ❞❡ 3Sk✳ P♦rt❛♥t♦✱
3Sk = 2.3 + 7.32+ 12.33+ 17.34+...+ 497.3100+ 502.3101.
▲♦❣♦✱
2Sk = 3Sk−Sk
= (502.3101−2)−5(3 + 32+ 33+ 34+...+ 3100)
= (502.3101−2)−5
2(3
101
−3),
✉t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❛ ♣r♦♣♦s✐çã♦ ✶✳✺✭❜✮ ❡ ❝❛❧❝✉❧❛♥❞♦✱ t❡♠♦s
1
4(999.3
101+ 11).
⋄
