Aplicações do método de indução matemática à geometria

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(1)❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ▼❛r❛♥❤ã♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❘❡❞❡ ✕ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧. ❘❆■▼❯◆❉❖ ❉❖ ◆❆❙❈■▼❊◆❚❖ ❱❊▲❖❩❖ ◆❊❚❖. ❆♣❧✐❝❛çõ❡s ❞♦ ▼ét♦❞♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ à ●❡♦♠❡tr✐❛. ❙ã♦ ▲✉ís ✷✵✶✼.

(2) ❘❆■▼❯◆❉❖ ❉❖ ◆❆❙❈■▼❊◆❚❖ ❱❊▲❖❩❖ ◆❊❚❖. ❆♣❧✐❝❛çõ❡s ❞♦ ▼ét♦❞♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ à ●❡♦♠❡tr✐❛ ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛✲ ❞✉❛çã♦ ❡♠ ❘❡❞❡ ✕ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ❞❛ ❯❋▼❆✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳. ❖r✐❡♥t❛❞♦r✿ Pr♦❢❛✳ ❘❡♥❛t❛ ❞❡ ❋❛r✐❛s ▲✐♠❡✐r❛ ❈❛r✈❛❧❤♦. ❉♦✉t♦r❛ ❡♠ ▼❛t❡♠át✐❝❛. ❙ã♦ ▲✉ís ✷✵✶✼.

(3) Ficha gerada por meio do SIGAA/Biblioteca com dados fornecidos pelo(a) autor(a). Núcleo Integrado de Bibliotecas/UFMA. Velozo Neto, Raimundo do Nascimento. Aplicações do Método de Indução Matemática à Geometria / Raimundo do Nascimento Velozo Neto. - 2017. 62 f. Orientador(a): Renata de Farias Limeira Carvalho. Dissertação (Mestrado) - Programa de Pós-graduação em Rede - Matemática em Rede Nacional/ccet, Universidade Federal do Maranhão, São Luís, 2017. 1. Demonstrações Matemáticas. 2. Geometria. 3. Método de indução matemática. I. Carvalho, Renata de Farias Limeira. II. Título..

(4) ➚ ♠✐♥❤❛ ♠ã❡✱ ▼❛r✐❛ ❞❛ ❈♦♥❝❡✐çã♦ ❙❛♥t♦s ❱❡❧♦③♦✳ ❆♦ ♠❡✉ ✜❧❤♦✱ ❇r✉♥♦ ❞❡ ▼♦r❛❡s ❱❡❧♦③♦✳ ❉❡❞✐❝♦✳.

(5) ❘❡s✉♠♦ ❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞♦ ▼ét♦❞♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❞❡ s❡✉ ✉s♦ ❝♦♠ ✈✐st❛s à s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s✳ ■♥✐❝✐❛❧♠❡♥t❡✱ sã♦ ❢❡✐t❛s ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s ❛❝❡r❝❛ ❞❛ ❡①♣r❡ssã♦ ✏r❛❝✐♦❝í♥✐♦ ✐♥❞✉t✐✈♦✑✱ ❝✉❥♦ s❡♥t✐❞♦✱ ❝♦♥❢♦r♠❡ ❛♣r♦♣r✐❛❞❛♠❡♥t❡ ❡①♣❧✐❝❛❞♦ ♥♦ t❡①t♦✱ ❞✐❢❡r❡ ❞♦ ❞❡ ✏✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✑✳ ➱ ♣r♦✈❛❞❛ ❛ ♣r♦♣♦s✐çã♦ q✉❡ ❣❛r❛♥t❡ ♦ ✉s♦ ❞♦ ♠ét♦❞♦ ❝♦♠ ❜❛s❡ ❡♠ s❡✉ ❢✉♥❞❛♠❡♥t♦✱ ❛ s❛❜❡r✱ ♦ ❛①✐♦♠❛ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛ ✭✉♠ ❞♦s ♣♦st✉❧❛❞♦s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✮ ❡ ❡①✐❜✐❞♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ s✉❛ ✉t✐❧✐③❛çã♦ ❡♠ ➪❧❣❡❜r❛ ❡ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✳ ❊♠ s❡❣✉✐❞❛✱ sã♦ ❡①♣❧♦r❛❞❛s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞♦ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛ à ♣r♦❜❧❡♠❛s ❞❡ ●❡♦♠❡tr✐❛✱ s❡❥❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❣❡♦♠étr✐❝❛ ❡♠ t❡r♠♦s ❞❡ ♦✉tr❛✭s✮✱ ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✉♠❛ ♣r♦♣♦s✐çã♦ q✉❡ s❡ ✐♥s✐♥✉❛ ✈❡r❞❛❞❡✐r❛✱ ♦✉ ♣❛r❛ ❛ ❡①✐❜✐çã♦ ❞❛s ❡t❛♣❛s ❞❡ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ ❞❛❞❛ ✜❣✉r❛✳. P❛❧❛✈r❛s✲❝❤❛✈❡✿ ▼ét♦❞♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✳ ❉❡♠♦♥str❛çõ❡s ♠❛t❡♠át✐❝❛s✳ Pr♦❜❧❡✲ ♠❛s ❞❡ ●❡♦♠❡tr✐❛✳.

(6) ❆❜str❛❝t ❚❤✐s ✇♦r❦ ❞❡❛❧s ✇✐t❤ t❤❡ ▼❡t❤♦❞ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ■♥❞✉❝t✐♦♥✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✐ts ✉s❡ ✇✐t❤ ❛ ✈✐❡✇ t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❣❡♦♠❡tr✐❝ ♣r♦❜❧❡♠s✳ ■t ✐♥✐t✐❛❧❧② s♦♠❡ ❝♦♥s✐❞❡r❛t✐♦♥s ❛r❡ ♠❛❞❡ ❛❜♦✉t t❤❡ ❡①♣r❡ss✐♦♥ ✏✐♥❞✉❝t✐✈❡ r❡❛s♦♥✐♥❣✑ ✇❤♦s❡ ✐t ♠❡❛♥✐♥❣✱ ❛s ❛♣♣r♦♣r✐❛t❡❧② ♠✉st ❜❡ ❡①♣❧❛✐♥❡❞ ✐♥ t❤❡ t❡①t✱ t❤❛t ❞✐✛❡rs ❢r♦♠ t❤❛t ♦❢ ✏♠❛t❤❡♠❛t✐❝❛❧ ✐♥❞✉❝t✐♦♥✑✳ ❲❡ ♣r♦✈❡ t❤❡ ♣r♦♣♦s✐t✐♦♥ t❤❛t ❣✉❛r❛♥t❡❡s t❤❡ ✉s❡ ♦❢ t❤❡ ♠❡t❤♦❞ ❜❛s❡❞ ♦♥ ✐ts ❢♦✉♥❞❛t✐♦♥✱ ♥❛♠❡❧② t❤❡ ❛①✐♦♠ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ✐♥❞✉❝t✐♦♥ ✭♦♥❡ ♦❢ t❤❡ ♣♦st✉❧❛t❡s t❤❛t ❝❤❛r❛❝t❡r✐③❡ t❤❡ ♥❛t✉r❛❧ ♥✉♠❜❡rs✮✳ ■t ❡①❤✐❜✐t❡❞ s♦♠❡ ❡①❛♠♣❧❡s ♦❢ ✐ts ✉s❡ ♦❢ ❆❧❣❡❜r❛ ❛♥❞ t❤❡ ❚❤❡♦r② ♦❢ ◆✉♠❜❡rs✳ ❆♥❞ t❤❡♥✱ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ♠❡t❤♦❞ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ✐♥❞✉❝t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠s ♦❢ ●❡♦♠❡tr② ❛r❡ ❡①♣❧♦r❡❞ t♦ ♦❜t❛✐♥ ❛ ❣❡♦♠❡tr✐❝ ♠❡❛s✉r❡ ✐♥ t❡r♠s ♦❢ ❛♥♦t❤❡r✭s✮✱ ❡✐t❤❡r ❢♦r t❤❡ ❞❡♠♦♥str❛t✐♦♥ ♦❢ ❛ ♣r♦♣♦s✐t✐♦♥ t❤❛t ✐♥s✐♥✉❛t❡s ✐ts❡❧❢ tr✉❡✱ ♦r ❢♦r t❤❡ st❛❣❡s ♦❢ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ✜❣✉r❡ ❣✐✈❡♥✳ ❑❡②✇♦r❞s✿. ▼❡t❤♦❞ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ✐♥❞✉❝t✐♦♥✳. ❜❧❡♠s ✐♥ ●❡♦♠❡tr②✳. ▼❛t❤❡♠❛t✐❝❛❧ ❞❡♠♦♥str❛t✐♦♥s✳. Pr♦✲.

(7) ❆❣r❛❞❡❝✐♠❡♥t♦s ❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ à ♠✐♥❤❛ ♠ã❡✱ ❉♦♥❛ ▼❛r✐❛ ❞❛ ❈♦♥❝❡✐çã♦✱ ♣♦r t❡r ❞❡❞✐❝❛❞♦✱ ❝♦♠ t♦❞♦ ❝❛r✐♥❤♦ ❡ ❣❡♥✉í♥♦ ❛♠♦r ❞❡ ♠ã❡✱ ♦s ♠❡❧❤♦r❡s ❛♥♦s ❞❡ s✉❛ ✈✐❞❛ à ❝✉✐❞❛r ❞❡ ♠❡✉s ✐r♠ã♦s ❡ ❞❡ ♠✐♠✳ ❊❧❛ é ♠❡✉ ♠❛✐♦r ❡①❡♠♣❧♦ ❞❡ ❞❡t❡r♠✐♥❛çã♦✱ ❤♦♥r❛❞❡③ ❡ ❛♠♦r à ✈✐❞❛✳ ❆♦ ♠❡✉ ✜❧❤♦✱ ❇r✉♥♦✱ q✉❡✱ ❝♦♠ s✉❛ ♥❛t✉r❛❧ ✐♥♦❝ê♥❝✐❛ ❞❡ ❝r✐❛♥ç❛✱ ✐♥s♣✐r❛✲♠❡ ❛ s❡r ❛❧❣✉é♠ ♠❡❧❤♦r ❡ ♠❡ ❢♦r♥❡❝❡ ♦ ✏❝♦♠❜✉stí✈❡❧✑ ♥❡❝❡ssár✐♦ ♣❛r❛ s❡❣✉✐r ❛❞✐❛♥t❡✳ ❆♦s ♠❡✉s ✐r♠ã♦s✱ ❈❛r❧❛ ❡ ❙❡❣✉♥❞♦✱ ❝✉❥❛ ❝♦♥✈✐✈ê♥❝✐❛ ❛♦ ❧♦♥❣♦ ❞♦s ❛♥♦s ♠✉✐t♦ ❝♦♥tr✐❜✉✐✉ ❡♠ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❝♦♠♦ s❡r ❤✉♠❛♥♦✳ ➚ ▲✐❞✐❛♥❡ ❉✉r❛♥s✱ ♣♦r t❡r ♠❡ ❞❛❞♦ ♦ q✉❡ t❡♥❤♦ ❞❡ ♠❛✐s ✐♠♣♦rt❛♥t❡ ♥❡st❛ ✈✐❞❛ ❡ ♣❡❧❛ ❣r❛♥❞❡ ♠ã❡ q✉❡ é✱ tr❛♥q✉✐❧✐③❛♥❞♦✲♠❡ ♥♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ♥ã♦ ♣♦ss♦ ❡st❛r ✜s✐❝❛♠❡♥t❡ ♣r❡s❡♥t❡ ❝♦♠ ♥♦ss♦ ✜❧❤♦✳ ➚ ❡st✐♠❛❞❛ ♣r♦❢❡ss♦r❛ ❘❡♥❛t❛ ❞❡ ❋❛r✐❛s ▲✐♠❡✐r❛ ❈❛r✈❛❧❤♦✱ ♣❡❧❛ ❣❡♥t✐❧❡③❛ ❡♠ ❛❝❡✐t❛r s❡r ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ ❡ ♣❡❧❛ ♣r❡st❛t✐✈❛✱ s❡❣✉r❛ ❡ ✈❛❧✐♦s❛ ♦r✐❡♥t❛çã♦ ❡❢❡t✐✈❛♠❡♥t❡ ❞❛❞❛✳ ➚s ♣r♦❢❡ss♦r❛s ❙❛♥❞r❛ ■♠❛❝✉❧❛❞❛ ▼♦r❡✐r❛ ◆❡t♦ ❡ ❱❛❧❞✐❛♥❡ ❙❛❧❡s ❆r❛ú❥♦✱ ♣❡❧❛ ❤♦♥r♦s❛ ♣r❡s❡♥ç❛ ♥❛ ❜❛♥❝❛ ❞❡ ❞❡❢❡s❛ ❞❡st❡ tr❛❜❛❧❤♦ ❡ ♣❡❧❛s ❝♦♥s✐❞❡r❛çõ❡s r❡❛❧✐③❛❞❛s ❛♦ ❝❛❜♦ ❞❡st❛✱ ♦ q✉❡✱ ❝❡rt❛♠❡♥t❡✱ ❞❡✉ ❛♦ t❡①t♦ ❡s❝r✐t♦ ♠❛✐s ✢✉✐❞❡③✱ ❝❧❛r❡③❛ ❡ ♣r❡❝✐sã♦✳ ❆♦s ♣r♦❢❡ss♦r❡s ◆❡r✐ ❚❡r❡③✐♥❤❛ ❇♦t❤ ❈❛r✈❛❧❤♦✱ ❏♦sé ▲✉✐③ ❘♦s❛s P✐♥❤♦ ❡ ❘✉❜❡♥s ❙t❛r❦❡✱ ❞❛ ❯❋❙❈✱ ♣❡❧♦s ❡①❡♠♣❧♦s ❞❡ ♣r♦✜ss✐♦♥❛❧✐s♠♦✱ s❡r✐❡❞❛❞❡ ❡ ❝♦♠♣❡tê♥❝✐❛ ❝♦♠ q✉❡ ❢♦♠♦s ❜r✐♥❞❛❞♦s✱ ♠❡✉s ❝♦❧❡❣❛s ❡ ❡✉✱ ❞✉r❛♥t❡ ♦ ♣❡rí♦❞♦ ❡♠ q✉❡ ❝✉rs❛♠♦s ❛ ▲✐❝❡♥❝✐❛t✉r❛ ❡♠ ▼❛t❡♠át✐❝❛✳ ❆♦s ❝♦❧❡❣❛s ❞❡ ♠❡str❛❞♦ ❞❛s t✉r♠❛s P❘❖❋▼❆❚ ✷✵✶✸ ❡ ✷✵✶✻✱ ❞❛ ❯❋▼❆✱ ♣❡❧❛ ❝♦♥✈✐✈ê♥❝✐❛ ❡ tr♦❝❛ ❞❡ ❡①♣❡r✐ê♥❝✐❛s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ❛♠✐❣♦ ▼❛r❝❡❧♦ ❖♠❛r ▲✐♠❛ ❞❡ ❈❛str♦✱ ♣❡❧❛s ❝♦♥✈❡rs❛s ✭♥❛s q✉❛✐s s❡♠♣r❡ ♠❡ ❜❡♥❡✜❝✐♦ ❞❡ s✉❛ ❡①♣❡r✐ê♥❝✐❛✮ ❡ ♣❡❧♦ ❡①❡♠♣❧♦ ❞❡ ❝✐❞❛❞ã♦ q✉❡ é✳ ❆♦ ❛♠✐❣♦ ❲❛❣♥❡r ❞❡ ❏❡s✉s P❡r❡✐r❛ ❙á✱ ♣❡❧♦s ❝♦♥st❛♥t❡s ✐♥❝❡♥t✐✈♦ ❡ ❝♦♥✜❛♥ç❛ q✉❡✱ ❣❡♥❡r♦s❛♠❡♥t❡✱ r❡❝❡❜♦✳.

(8) ❆♦s ❝r✐❛❞♦r❡s ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ♦ P❘❖❋▼❆❚✱ ✉♠ ♣r♦❥❡t♦ q✉❡ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠❛ ❛♠♣❧✐❛çã♦ ❡♠ ❧❛r❣❛ ❡s❝❛❧❛ ❞♦ Pr♦❣r❛♠❛ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ♣❛r❛ Pr♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♦ P❆P✲ ▼❊▼✱ ❛ ❝✉❥♦ ✐❞❡❛❧✐③❛❞♦r✱ ♦ s❛✉❞♦s♦ ♣r♦❢❡ss♦r ❊❧♦♥ ▲❛❣❡s ▲✐♠❛✱ r❡s♣❡✐t♦s❛ ❡ ✈✐✈❛♠❡♥t❡✱ ♠❡♥❝✐♦♥♦ ❡ ❛❣r❛❞❡ç♦✳ ❆♦s ❣❡st♦r❡s ❞♦ ❈❊ ❉❛②s❡ ●❛❧✈ã♦ ❞❡ ❙♦✉s❛✱ ♣r♦❢❡ss♦r❡s ▲✐❧✐❛♥ ❆ss✉♥çã♦ ❡ ❋❡r♥❛♥❞♦ ❋♦♥s❡❝❛✱ ♣❡❧♦ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ❝♦♠♣r❡❡♥sã♦ ❞❡♠♦♥str❛❞♦s ❞✉r❛♥t❡ ❛ ✏r❡t❛ ✜♥❛❧✑ ❞♦ ❝✉rs♦ ✭❡♠ ♣❛rt✐❝✉❧❛r✱ ♥♦ ♣❡rí♦❞♦ ❞♦s ❡st✉❞♦s ❝♦♠ ✈✐st❛s à ♣r♦❞✉çã♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✮✳ ➚ ❈❆P❊❙✱ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳.

(9) ✏❯♠ ❤♦♠❡♠ ❞❡ ♠♦r❛❧✳✳✳ ✳✳✳r❡❝♦♥❤❡❝❡ ❛ q✉❡❞❛ ❡ ♥ã♦ ❞❡s❛♥✐♠❛✳ ▲❡✈❛♥t❛✱ s❛❝♦❞❡ ❛ ♣♦❡✐r❛ ❡ ❞á ❛ ✈♦❧t❛ ♣♦r ❝✐♠❛✳✑ P❛✉❧♦ ❱❛♥③♦❧✐♥✐ ✭❝✐❡♥t✐st❛ ❡ ❝♦♠♣♦s✐t♦r✮.

(10) ❙✉♠ár✐♦ ▲✐st❛ ❞❡ ❋✐❣✉r❛s. ✽. ■♥tr♦❞✉çã♦. ✽. ✶ ❖ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛. ✶✷. ✶✳✶. ❘❛❝✐♦❝í♥✐♦ ✐♥❞✉t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✶✷. ✶✳✷. ■♥❞✉çã♦ ♠❛t❡♠át✐❝❛. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✶✻. ✶✳✷✳✶. ❊①❡♠♣❧♦ ❞❡ ❞❡♠♦♥str❛çã♦ ♣♦r ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✶✻. ✶✳✷✳✷. Pr✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✶✼. ❉❡♠♦♥str❛çõ❡s ♣♦r ✐♥❞✉çã♦ ❡♠ ➪❧❣❡❜r❛ ❡ ❡♠ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ✳ ✳ ✳ ✳ ✳. ✶✾. ✶✳✸✳✶. ■❞❡♥t✐❞❛❞❡s ❡ ♣r♦❜❧❡♠❛s ❛r✐t♠ét✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✷✵. ✶✳✸✳✷. ❉❡s✐❣✉❛❧❞❛❞❡s ❡ ✭♦✉tr♦s✮ ♣r♦❜❧❡♠❛s ❛❧❣é❜r✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✷✸. ✶✳✸. ✷ ❆♣❧✐❝❛çõ❡s ❞❛ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛ à ♣r♦❜❧❡♠❛s ❞❡ ●❡♦♠❡tr✐❛. ✷✻. ✷✳✶. ❈á❧❝✉❧♦ ♣♦r ✐♥❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✷✻. ✷✳✷. ❉❡♠♦♥str❛çã♦ ♣♦r ✐♥❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✸✺. ✷✳✸. ❈♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s ♣♦r ✐♥❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✹✸. ✸ ❈♦♥s✐❞❡r❛çõ❡s. ✺✶. ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s. ✺✸. ❆ ❉❡♠♦♥str❛çõ❡s s❡❧❡❝✐♦♥❛❞❛s. ✺✺. ❆✳✶ ❊①♣r❡ssã♦ ❞❛ ❞✉♣❧✐❝❛çã♦ ❞♦ ❧❛❞♦ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❞❡. 2n. ❧❛❞♦s ✳ ✳ ✳ ✳. ❆✳✷ ❊①♣r❡ssã♦ ❞❛ t❛♥❣❡♥t❡ ❞♦ ❛r❝♦ ♠❡t❛❞❡ ❡♠ ❢✉♥çã♦ ❞♦s ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦. ✳. ✺✺ ✺✻.

(11) ▲✐st❛ ❞❡ ❋✐❣✉r❛s ✶✳✶. ■❧✉str❛çã♦ ❞❛ ✐❞é✐❛ ❞❛ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛. ✷✳✶. P♦❧í❣♦♥♦ r❡❣✉❧❛r ❞❡. ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✸✵. ✷✳✷. ◗✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✸✸. ✷✳✸. ◗✉❛❞r✐❧át❡r♦ ♥ã♦✲❝♦♥✈❡①♦. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✸✸. ✷✳✹. ❙❡♠✐rr❡t❛s t❡r♠✐♥❛♥❞♦ ❡♠ ✉♠ ♠❡s♠♦ ❧❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✸✹. ✷✳✺. ❙❡♠✐rr❡t❛s ✏❛t✐♥❣✐♥❞♦✑ ♠❛✐s ❞♦ q✉❡ ✶ ❧❛❞♦. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✸✹. ✷✳✻. ◗✉❛❞r❛❞♦s. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✸✺. ✷✳✼. ◗✉❛❞r❛❞♦. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✸✻. ✷✳✽. ❚r✐â♥❣✉❧♦s. ABCD. 2n. ❧❛❞♦s✱ ❞❡ ♣❡rí♠❡tr♦. A1 B1 C1 D1. ❡. A2 B2 C2 D2. ACM1 , M1 CM2 , . . . , Mn−1 CB. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. P✱. ❡ s✉❛ ❝♦r❞❛. AB. ❡ s✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ✐♥s❝r✐t❛s ❡. ❡①✐♥s❝r✐t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✾. ❈✐r❝✉♥❢❡rê♥❝✐❛ ❡①✐♥s❝r✐t❛ ♥♦ tr✐â♥❣✉❧♦. ✷✳✶✵ ❚r✐â♥❣✉❧♦. ABC. ✸✽. ✉s❛❞♦ ❝♦♠♦ ❛✉①✐❧✐❛r ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❝♦r♦❧ár✐♦ ✷✳✷✳✶ ✳ ✳. ✸✾. C1 , . . . , Cn ✱. ✷✳✶✷ ❈✐r❝✉♥❢❡rê♥❝✐❛s. C1. ✷✳✶✹ ◗✉❛❞r✐❧át❡r♦ ✷✳✶✺ ❚r✐â♥❣✉❧♦. ✹✶. C2 ✱. ❛✉①✐❧✐❛r❡s ♥❛ ♣r♦✈❛ ❞♦ ▲❡♠❛ ✷✳✷✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✹✶. OB1 A3 B3 ✑. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✹✷. X1 X2n−1 X2n X2n+1. n. ❧❛❞♦s. ✷✳✶✼ ❘❡t❛s ♣❛r❛❧❡❧❛s ✷✳✶✽. Tn. ❡. Qn ✱. ❡. r1. ❡. X1 X2 . . . Xn r2. ❜❛s❡ ❞❛ ✐♥❞✉çã♦. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. A1 , . . . , A n , A. ✹✺ ✹✼. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✹✼. ′. ❛ ♣❛rt✐r ❞♦s ✈ért✐❝❡s. ✹✹. ✳ ✳ ✳ ✳. ✐♥t❡rs❡❝çõ❡s ❞❛s r❡t❛s. P♦❧í❣♦♥♦ r❡❣✉❧❛r. ♣❛ss❛♥❞♦ ♣♦r ✉♠ ♠❡s♠♦ ♣♦♥t♦. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. n = 3✱. ❞♦ ❝❛s♦. ✷✳✶✾ ❉❡t❡r♠✐♥❛çã♦ ❞♦s ♣♦♥t♦s. ❆✳✶. ❝♦♠. n > 3✱. ✳ ✳. X1 X2 X3 ✱. ✷✳✶✻ P♦❧í❣♦♥♦ ❞❡. ABC. ✸✼. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✷✳✶✶ ❈✐r❝✉♥❢❡rê♥❝✐❛s. ✷✳✶✸ ✏◗✉❛❞r✐❧át❡r♦. ✶✽. AD ′′. ❡. r2. Pn+1 , Pn+1 , . . .. A1 A2 . . . A2n ✱. ❝♦♠ ❛ r❡t❛. XPn. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✹✽. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳. ✺✵. ✐♥s❝r✐t♦ ❡♠ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦. R. ✳ ✳. ✺✺.

(12) ✾. ■♥tr♦❞✉çã♦ ❈♦♠♦ s❡✉ ♣ró♣r✐♦ tít✉❧♦ ✐♥❞✐❝❛✱ ❡st❛ ❞✐ss❡rt❛çã♦ ❛❜♦r❞❛ ♦ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✱ ❝♦♠ ê♥❢❛s❡ ❡♠ s✉❛s ❛♣❧✐❝❛çõ❡s à ●❡♦♠❡tr✐❛✳ ❆ ♣r✐♥❝✐♣❛❧ r❛③ã♦ ♣❡❧❛ q✉❛❧ ❡s❝♦❧❤❡♠♦s ❡st❡ t❡♠❛ ♣❛r❛ ❡st✉❞♦ ❢♦✐ ♥♦ss♦ ❡♥t✉s✐✲ ❛s♠♦ ♣❡❧❛s ❝❤❛♠❛❞❛s ❞❡♠♦♥str❛çõ❡s. ♠❛t❡♠át✐❝❛s. ✶. ✱ ❡st❡s ❛t♦s ❞❡ ♣❡rs✉❛sã♦ ❢✉♥❞❛♠❡♥t❛✲. ❞♦s ❡♠ ❛r❣✉♠❡♥t❛çõ❡s ❧ó❣✐❝♦✲❞❡❞✉t✐✈❛s✱ ♣♦r ♠❡✐♦ ❞♦s q✉❛✐s ♥♦s ❝♦♥✈❡♥❝❡♠♦s ❞❡ q✉❡ ✉♠ ❢❛t♦ ♠❛t❡♠át✐❝♦ ✭✉♠❛ ♣r♦♣♦s✐çã♦✱ ✉♠ ❧❡♠❛✱ ✉♠ t❡♦r❡♠❛✱ ❡t❝✳✮ é ✈á❧✐❞♦✳ ❯♠ ❡♣✐só❞✐♦ ♦❝♦rr✐❞♦ q✉❛♥❞♦ ❛✐♥❞❛ ❝✉rs❛✈❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦ ♠❡ ❞❡s♣❡rt♦✉ ❛ ❝✉r✐♦s✐❞❛❞❡ ❡ ♦ ✐♥t❡r❡ss❡ ♣❡❧❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❥✉st✐✜❝❛r ✉♠❛ ❛✜r♠❛çã♦ ♠❛t❡♠át✐❝❛ ✭♣♦r ♠❛✐s ó❜✈✐❛ q✉❡ ❡st❛ ♣♦ss❛ ♣❛r❡❝❡r✮✳ ■♥❞❛❣❛❞♦ s♦❜r❡ ♦ ♣♦rq✉ê ❞❡ ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ s❡r s✉❜❝♦♥❥✉♥t♦ ❞❡ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦✱ ♠❡✉ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ à ❛❧t✉r❛ r❡s♣♦♥❞❡✉✲♠❡✿ ✏❊♠❜♦r❛ ♥ã♦ s❡❥❛ ❡①❛t❛♠❡♥t❡ ❝♦♠♦ ❧❤❡ ❞✐❣♦✱ ✈♦❝ê ♣♦❞❡ s❡ ❝♦♥✈❡♥❝❡r ❞❛ ✈❡r❞❛❞❡ ❞❡st❡ ❢❛t♦ ❜❛st❛♥❞♦ ✐♠❛❣✐♥❛r q✉❡✱ s❡ ❡❧✐♠✐♥áss❡♠♦s ❞❡st❡ ❝♦♥❥✉♥t♦ t♦❞♦s ♦s s❡✉s ❡❧❡♠❡♥t♦s✱ ♦ q✉❡ ✏s♦❜r❛r✐❛✑ s❡r✐❛ ♦ ✈❛③✐♦ ❡♠ s❡✉ ✐♥t❡r✐♦r✳✑. ❆♣❡s❛r ❞❡✱ ♥❛q✉❡❧❛ ♦❝❛s✐ã♦✱ ✐st♦ t❡r ❢❡✐t♦. ❛❧❣✉♠ s❡♥t✐❞♦ ♣❛r❛ ♠✐♠✱ ♣❛ss❡✐ ❛❧❣✉♠ t❡♠♣♦ q✉❡st✐♦♥❛♥❞♦✲♠❡ s♦❜r❡ ❛ ♥❛t✉r❡③❛ ❞❡st❛ ♣r♦✈❛✳✳✳. ❍♦❥❡✱ ❡♠❜♦r❛ r❡❝♦♥❤❡ç❛ ♥❡st❛ ♣❛ss❛❣❡♠ ✉♠ ❛t♦ ❞❡ ❢r❛♥q✉❡③❛ ♣♦r ♣❛rt❡ ❞❡ ♠❡✉. ❛♥t✐❣♦ ♠❡str❡✱ ♣r❡♦❝✉♣❛❞♦ ❡♠ ❡s❝❧❛r❡❝❡r ❛ ❞ú✈✐❞❛ q✉❡ ♦ ❧❛♥ç❛✈❛ ❡✱ ❛✐♥❞❛✱ ❞❡ ♠♦❞♦ ❛ s❡r ❝♦♠♣r❡❡♥❞✐❞♦✱ t❡♥❤♦ ❝♦♥✈✐❝çã♦ ❞❡ q✉❡ ✉♠❛ ❞❡♠♦♥str❛çã♦ ♣❛r❛ ♦ ❢❛t♦ ❞❡ q✉❡ ∅ ⊂ A✱. q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ❝♦♥❥✉♥t♦ A✱ ❢❡✐t❛ ♣♦r r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦✱ t❡r✐❛ s✐❞♦✱ ♣♦r ♠✐♠✱ ♥❛q✉❡❧❡ ♠♦♠❡♥t♦✱ ♣❧❡♥❛♠❡♥t❡ ❝♦♠♣r❡❡♥❞✐❞❛ ❡✱ ♣♦r ✐ss♦ ♠❡s♠♦✱ ❛❝❡✐t❛✳ ❆❧é♠ ❞♦ ♠❡♥❝✐♦♥❛❞♦ ❛❝✐♠❛✱ ❡♥t❡♥❞♦ s❡r ❝♦♥✈❡♥✐❡♥t❡ ❝✐t❛r ❛ ❧❡✐t✉r❛ q✉❡ ✜③ ❞♦ ❛rt✐❣♦ ✐♥t✐t✉❧❛❞♦. ✏❉❡❝♦r❛r é ♣r❡❝✐s♦✱ ❞❡♠♦♥str❛r t❛♠❜é♠ é✑ ✱. ●❛r❜✐✷ ✱ ♣✉❜❧✐❝❛❞♦ ❤á ❛❧❣✉♥s ❛♥♦s ♥❛. ❞❡ ❛✉t♦r✐❛ ❞❡ ●✐❧❜❡rt♦. ❘❡✈✐st❛ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✳. ◆❡❧❡✱ ♦ ❛✉t♦r. ✶ ✏◆❡♥❤✉♠❛ ✐♥✈❡st✐❣❛çã♦ ❢❡✐t❛ ♣❡❧♦ ❤♦♠❡♠ ♣♦❞❡ s❡r ❝❤❛♠❛❞❛ r❡❛❧♠❡♥t❡ ❞❡ ❝✐ê♥❝✐❛ s❡ ♥ã♦ ♣✉❞❡r s❡r ❞❡♠♦♥str❛❞❛ ♠❛t❡♠❛t✐❝❛♠❡♥t❡✳✑ ✭❋r❛s❡ ❞❡✈✐❞❛ ❛♦ ♠❛t❡♠át✐❝♦✱ ❛♥❛t♦♠✐st❛✱ ❜♦tâ♥✐❝♦✱ ✐♥✈❡♥t♦r✱ ❝✐❡♥t✐st❛✱ ♣♦❡t❛ ❡ ❡s❝✉❧t♦r✱ ▲❡♦♥❛r❞♦ ❞❛ ❱✐♥❝✐ ❬✶✹✺✷✕✶✺✶✾❪✱ ✉♠ ❞♦s ❤♦♠❡♥s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞♦ ❘❡♥❛s❝✐♠❡♥t♦✳✮. ✷ ●✐❧❜❡rt♦ ●✳ ●❛r❜✐ é ❡♥❣❡♥❤❡✐r♦ ❡ ❛✉t♦r ❞♦s ❧✐✈r♦s ❆ ❘❛✐♥❤❛ ❞❛s ❈✐ê♥❝✐❛s ❡ ❖ ❘♦♠❛♥❝❡ ❞❛s ❊q✉❛çõ❡s ❆❧❣é❜r✐❝❛s✱ ♣✉❜❧✐❝❛❞♦s ♣❡❧❛ ❡❞✐t♦r❛ ▲✐✈r❛r✐❛ ❞❛ ❋ís✐❝❛✳ ❙❡✉ ❛rt✐❣♦ ❝✐t❛❞♦ ♥♦ t❡①t♦ ♣♦❞❡ s❡r ❝♦♥s✉❧t❛❞♦ ♥❛ ♣á❣✐♥❛ ❡❧❡trô♥✐❝❛ ❞❛ ❘❡✈✐st❛ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ❜❛st❛♥❞♦ ♣❛r❛ ✐ss♦ ❛❝❡ss❛r ♦ ❝❛♠♣♦ ✏❆rt✐❣♦s P✉❜❧✐❝❛❞♦s✑✳.

(13) ✶✵ tr❛t❛ ❞♦ ✭❢❛❧s♦✮ ❞✐❧❡♠❛ ❡♥tr❡ ❝♦♠♣r❡❡♥❞❡r ❡ ♠❡♠♦r✐③❛r ✭♦✉ ❞❡❝♦r❛r ❡ ❡♥t❡♥❞❡r✮✱ ❛❧❡rt❛♥❞♦✲ ♥♦s ♣❛r❛ ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ✐❞❡❛❧ s❡r✐❛ q✉❡ ❛ ❝♦♠♣r❡❡♥sã♦ ♣r❡❝❡❞❡ss❡ ❛ ♠❡♠♦r✐③❛çã♦✱ ✉♠❛ ♥ã♦ ❡①❝❧✉✐♥❞♦ ❛ ♦✉tr❛✳ ❊①♣õ❡ ♦ ❛✉t♦r✱ ❛✐♥❞❛✱ ❛s r❛③õ❡s ♣❡❧❛s q✉❛✐s ❝♦♥s✐❞❡r❛ ♣r❡❥✉❞✐❝✐❛❧ ♦ ❡♥❢♦q✉❡ ✉t✐❧✐t❛r✐st❛ q✉❡✱ ♣♦r ✈❡③❡s✱ é ❞❛❞♦ à ▼❛t❡♠át✐❝❛✱ ♦ q✉❡ ❛❝❛❜❛ ♣♦r ❧✐♠✐t❛r ♦ r❛❝✐♦❝í♥✐♦ ❡✱ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ♣♦r ❞✐✜❝✉❧t❛r ❛ ❛q✉✐s✐çã♦ ❞❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♣❡♥s❛r ❞❡ ❢♦r♠❛ ❣❡♥ér✐❝❛ ❡ ❛❜str❛t❛✱ ❛❧❣♦ tã♦ ✐♠♣♦rt❛♥t❡ às ♣❡ss♦❛s r❡❛❧♠❡♥t❡ ❝✉❧t❛s✳ P♦rt❛♥t♦✱ ❝❤❡❣❛❞❛ ❛ ❤♦r❛ ❞❡ ❞❡❝✐❞✐r s♦❜r❡ ♦ t❡♠❛ ❞❡st❛ ❞✐ss❡rt❛çã♦✱ ❞ú✈✐❞❛s ♥ã♦ ❤♦✉✈❡r❛♠✳ ❊♠ ✈❡r❞❛❞❡✱ ❡r❛ ❛♣❡♥❛s q✉❡stã♦ ❞❡ s❡ ❡s❝♦❧❤❡r ❛ ✏♠❡❧❤♦r r♦t❛✑ ❛ s❡❣✉✐r✱ ♥♦ q✉❡ ❛s ❞✐s❝✐♣❧✐♥❛s. ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ ❡ ❆r✐t♠ét✐❝❛ ✭❝✉rs❛❞❛s ♥♦s ♣r✐♠❡✐r♦s s❡♠❡str❡s. ❞♦ ♠❡str❛❞♦✮ ♠✉✐t♦ ❝♦♥tr✐❜✉ír❛♠✱ ♥❛ ♠❡❞✐❞❛ ❡♠ q✉❡ ♥♦s ❡①♣✉♥❤❛♠ ❛ ✐♠♣♦rtâ♥❝✐❛ ❡ ✉t✐❧✐❞❛❞❡ ❞♦ ♠ét♦❞♦ ❞❡ ❞❡♠♦♥str❛çã♦ ♣♦r ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛ ♥❛ ♣r♦✈❛ ❞❡ ♣r♦❜❧❡♠❛s ❛❧❣é❜r✐❝♦s ❡ ❛r✐t♠ét✐❝♦s✳ ❖s ❡①❡r❝í❝✐♦s ❡ ♣r♦❜❧❡♠❛s ❝♦❧✐❣✐❞♦s ♥❡st❡ tr❛❜❛❧❤♦ ❢♦r❛♠ ❡①tr❛í❞♦s✱ ❡♠ s✉❛. ▼ét♦❞♦ ❞❡ ■♥❞✉❝❝✐ó♥ ▼❛t❡♠át✐❝❛ ❡ ■♥❞✉❝❝✐ó♥ ❡♥ ❧❛ ●❡♦♠❡trí❛✱ ❞❛ sér✐❡ ▲❡❝❝✐♦♥❡s P♦♣✉❧❛r❡s ❞❡ ▼❛t❡♠át✐❝❛s✱ ♠❛✐♦r✐❛✱ ❞♦s ♣❡q✉❡♥♦s✱ ♣♦ré♠ ❞❡ ✐♥❡st✐♠á✈❡❧ ✈❛❧♦r ❡❞✉❝❛t✐✈♦✱ ❧✐✈r♦s. ❞❛ ❊❞✐t♦r❛ ▼✐r✳ ❖ s❡❣✉♥❞♦ ❞❡❧❡s ❢❛③✱ ❛✐♥❞❛✱ ❝♦♥❡①õ❡s ❝♦♠ ♦✉tr❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛ ✭❛❧é♠ ❞❛ ●❡♦♠❡tr✐❛✮✱ ❝♦♠♦ ❛. ❚❡♦r✐❛ ❞♦s ●r❛❢♦s ❡ ❛ ❈♦♠❜✐♥❛tór✐❛✱ ❛❜♦r❞❛♥❞♦ ♣r♦❜❧❡♠❛s. s♦❜r❡ ♠❛♣❛s ❣❡♦❣rá✜❝♦s ❡ ❜♦❛ ❝♦❧♦r❛çã♦ ❞❡ ♠❛♣❛s✱ ❛q✉✐ ♥ã♦ ❡①♣❧♦r❛❞♦s✳ ◆♦s ❡①❡r❝í❝✐♦s ✉t✐❧✐③♦ ♦ sí♠❜♦❧♦ ♦ sí♠❜♦❧♦. . . ♣❛r❛ ♠❛r❝❛r ♦ ✜♥❛❧ ❞❛ ❞❡♠♦♥str❛çã♦✱ ❝♦♥q✉❛♥t♦ q✉❡✱ ♥♦s ♣r♦❜❧❡♠❛s✱. é ✉s❛❞♦ ❛♦ tér♠✐♥♦ ❞❛ ♣r♦✈❛ ✭✐♥t✐t✉❧❛❞❛ ♣♦r. s♦❧✉çã♦ ✮✳. ❆ s❡❣✉✐r✱ é ❞❛❞❛ ✉♠❛ s✉❝✐♥t❛ ❞❡s❝r✐çã♦ ❞❡ ❝❛❞❛ ✉♠ ❞♦s ❞♦✐s ❝❛♣ít✉❧♦s ❡♠ q✉❡ ❡st❡ tr❛❜❛❧❤♦ ❡stá ❞✐✈✐❞✐❞♦✳ ❯♠ ❡①♣❧✐❝❛çã♦ ❛❝❡r❝❛ ❞❛s ❞✐❢❡r❡♥ç❛s ❡①✐st❡♥t❡s ❡♥tr❡ ♦s s✐❣♥✐✜❝❛❞♦s ❞❛s ❡①♣r❡s✲ sõ❡s. r❛❝✐♦❝í♥✐♦ ✐♥❞✉t✐✈♦ ❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✱ ❛❧é♠ ❞❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❞❡♠♦♥str❛çõ❡s. ❞❡ ♣r♦❜❧❡♠❛s ❛❧❣é❜r✐❝♦s ❡ ❛r✐t♠ét✐❝♦s✱ r❡❛❧✐③❛❞❛s ❝♦♠ ❜❛s❡ ♥♦ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡✲ ♠át✐❝❛✱ sã♦ ❢♦r♥❡❝✐❞♦s ♥♦ ❈❛♣ít✉❧♦ ✶✳ ❙❡✉ ♦❜❥❡t✐✈♦ é ✏♣r❡♣❛r❛r ♦ t❡rr❡♥♦✑ ♣❛r❛ ♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✱ q✉❡ tr❛t❛ ❞♦ ✉s♦ ❞❛ ✐♥❞✉çã♦ ♥❛ ●❡♦♠❡tr✐❛✳ ❖ ❈❛♣ít✉❧♦ ✷ é ♦ ❝❡r♥❡ ❞❡st❛ ❞✐ss❡rt❛çã♦✳ ◆❡❧❡✱ sã♦ ❡①✐❜✐❞❛s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞♦ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛ ❢❡✐t❛s ❡♠ ♣r♦❜❧❡♠❛s ❡①tr❛í❞♦s ❞❛ ●❡♦♠❡tr✐❛✳ ❞✐✈✐❞✐❞♦ ❡♠ três s❡çõ❡s✱ ♥❛s q✉❛✐s ♦s ♣r♦❜❧❡♠❛s ❡stã♦ ❛ss✐♠ ♥♦♠❡❛❞♦s✿. ❝á❧❝✉❧♦✱. ❞❡. ❞❡♠♦♥str❛çõ❡s. ❡ ❞❡. ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s✳. ❋♦✐. ♣r♦❜❧❡♠❛s ❞❡. P❛r❛ ❛❧é♠ ❞❡ ✉♠❛ ✏s✐♠♣❧❡s✑. ❝♦♠♣✐❧❛çã♦ ❞❡ ❡①❡r❝í❝✐♦s✲♣r♦❜❧❡♠❛✱ ❡st❡ ❝❛♣ít✉❧♦ t❛♠❜é♠ s❡ ♣r❡st❛ ❛ ❛♣r❡s❡♥t❛r ❞✐❢❡r❡♥t❡s.

(14) ✶✶ ♠♦❞♦s ❞❡ s❡ ❛♣❧✐❝❛r ♦ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠♦ ♦ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛ ❣❡♦♠étr✐❝♦ ❛❜♦r❞❛❞♦✳ ❆❧é♠ ❞✐ss♦✱ ♦s ♣ró♣r✐♦s ♣r♦❜❧❡♠❛s ✭❡♠ ❛❧❣✉♥s ❝❛s♦s✱ ♣❡❧♦ ♠❡♥♦s✮ ❥á ❤❛✈❡rã♦ ❞❡ ❞❡s♣❡rt❛r ✐♥t❡r❡ss❡ ♣♦r s✐ ♣ró♣r✐♦s✱ ♥❛ ♠❡❞✐❞❛ ❡♠ q✉❡ s♦❧✐❝✐t❛♠ ❞❡ q✉❡♠ ♣r❡t❡♥❞❛ r❡s♦❧✈ê✲❧♦s ✭❛♥t❡s ♠❡s♠♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞❛ ✐♥❞✉çã♦✮ ❜♦❛ ❞♦s❡ ❞❡ ❡♥❣❡♥❤♦s✐❞❛❞❡✳ ❊♠ ❛❧❣✉♥s ❞❡❧❡s ✭❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ♥♦s ❝♦r♦❧ár✐♦s✮✱ ❛❧✐ás✱ ♣❛r❛ s✉❛ r❡s♦❧✉çã♦✱ ♥ã♦ ✜③❡♠♦s ✉s♦ ❞✐r❡t♦ ❞♦ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦❀ ❛ ❞❡❝✐sã♦ ❞❡ ✐♥❝❧✉í✲❧♦s ♥♦ tr❛❜❛❧❤♦ é ❞❡✈✐❞❛ ❛♦ ❢❛t♦ ❞❡ q✉❡✱ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❜✉s❝❛ ❞❛ s♦❧✉çã♦✱ sã♦ ✉t✐❧✐③❛❞♦s r❡s✉❧t❛❞♦s q✉❡✱ ♣♦r s✉❛ ✈❡③✱ ❢♦r❛♠ ❞❡♠♦♥str❛❞♦s ♣♦r ✐♥❞✉çã♦✳ P♦r ✜♠✱ ❞❡✈❡ s❡r s❛❧✐❡♥t❛❞♦ q✉❡ ❡st❡ tr❛❜❛❧❤♦ ♣r❡t❡♥❞❡ s❡r ❝❛♣❛③ ❞❡ ③✐r. ✐♥❞✉✲. ♣r♦❢❡ss♦r❡s ❡ ♣r♦❢❡ss♦r❛s ✭❡ ❛ q✉❡♠ ♠❛✐s s❡ ♦❝✉♣❛ ❞❡ ▼❛t❡♠át✐❝❛✮ ❛ s❡ ❛♣r♦♣r✐❛r. ❞♦ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✱ ❝♦♠♣r❡❡♥❞❡♥❞♦ s✉❛ ❛t✉❛çã♦ ✐♥❞✐s♣❡♥sá✈❡❧ ♥❛ ♣r♦✈❛ ❞❡ ♣r♦♣♦s✐çõ❡s q✉❡✱ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ r❡❧❛❝✐♦♥❛♠ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❞♦♠✐♥❛♥❞♦ s✉❛s ❢❛❝❡t❛s ❡ s❡✉s ❞✐✈❡rs♦s ♠♦❞♦s ❞❡ ✉t✐❧✐③❛çã♦ ✭❡♠ ♣❛rt✐❝✉❧❛r✱ ❛q✉❡❧❡s ♠❛✐s ❧✐❣❛❞♦s à ●❡♦✲ ♠❡tr✐❛✱ ❝♦♠♦ ♦s q✉❡ ❛q✉✐ ❢♦r❛♠ ❡①♣❧♦r❛❞♦s✮ ❡✱ ❡♠ s❡♥❞♦ ♣♦ssí✈❡❧✱ ❝r✐❛♥❞♦ ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ♦s ❡st✉❞❛♥t❡s ✭❡♠ ❡s♣❡❝✐❛❧✱ ♦s ❞❛ ❡s❝♦❧❛ ❜ás✐❝❛ ❡ ❛q✉❡❧❡s ❞♦s ♣r✐♠❡✐r♦s ♣❡rí♦❞♦s ❞♦ ❡♥s✐♥♦ ✉♥✐✈❡rs✐tár✐♦✮✱ ❞❡s❞❡ ❧♦❣♦✱ s❡❥❛♠ ♣♦st♦s ❡♠ ❝♦♥t❛t♦ ❝♦♠ ❡st❛ ❡❧❡❣❛♥t❡ té❝♥✐❝❛ ❞❡ ❞❡♠♦♥str❛çã♦ ♠❛t❡♠át✐❝❛ ❡ ✐♠♣r❡ss✐♦♥❛♥t❡ ❢♦r♠❛ ❞❡ s❡✱ ♣♦r ❛ss✐♠ ❞✐③❡r✱ ❧✐❞❛r ❝♦♠ ♦ ✐♥✜♥✐t♦✦.

(15) ✶✷. ✶ ❖ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛ ✶✳✶. ❘❛❝✐♦❝í♥✐♦ ✐♥❞✉t✐✈♦. ❉❡♥♦♠✐♥❛✲s❡. ✐♥❞✉çã♦. ❛ t♦❞♦ r❛❝✐♦❝í♥✐♦ q✉❡ ❝♦♠♣r❡❡♥❞❡ ❛ ♣❛ss❛❣❡♠ ❞❡ ♣r♦♣♦s✐çõ❡s ♣❛r✲. t✐❝✉❧❛r❡s ❛ ♣r♦♣♦s✐çõ❡s ❣❡r❛✐s✱ ❝♦♠ ❛ ♣❡❝✉❧✐❛r✐❞❛❞❡ ❞❡ q✉❡ ❛ ✈❛❧✐❞❛❞❡ ❞❛s ú❧t✐♠❛s s❡ ❞❡✲ ♣r❡❡♥❞❡ ❞❛ ✈❛❧✐❞❛❞❡ ❞❛s ♣r✐♠❡✐r❛s✳ ❊①❡♠♣❧♦s ❞❡ ♣r♦♣♦s✐çõ❡s ✭♦✉ ❛✜r♠❛çõ❡s✮ ❣❡r❛✐s sã♦ ❛s s❡❣✉✐♥t❡s✿. ❼. ❚♦❞♦s sã♦ ✐❣✉❛✐s ♣❡r❛♥t❡ ❛ ❧❡✐✱ s❡♠ ❞✐st✐♥çã♦ ❞❡ q✉❛❧q✉❡r ♥❛t✉r❡③❛✱ ❣❛r❛♥t✐♥❞♦✲s❡ ❛♦s ❜r❛s✐❧❡✐r♦s ❡ ❛♦s ❡str❛♥❣❡✐r♦s r❡s✐❞❡♥t❡s ♥♦ P❛ís ❛ ✐♥✈✐♦❧❛❜✐❧✐❞❛❞❡ ❞♦ ❞✐r❡✐t♦ à ✈✐❞❛✱ à ❧✐❜❡r❞❛❞❡✱ à ✐❣✉❛❧❞❛❞❡✱ à s❡❣✉r❛♥ç❛ ❡ à ♣r♦♣r✐❡❞❛❞❡✱ ❬✳ ✳ ✳ ❪. ✭❈♦♥st✐t✉✐çã♦. ❋❡❞❡r❛❧✱ ❈❛♣ít✉❧♦ ■✱ ❛rt✐❣♦ ✺➸✮✳. ❼. ❊♠ t♦❞♦ tr✐â♥❣✉❧♦✱ ♦ ❜❛r✐❝❡♥tr♦ ✭♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛s três ♠❡❞✐❛♥❛s✮ ❞✐✈✐❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ♠❡❞✐❛♥❛s ❞❡ t❛❧ ♠♦❞♦ q✉❡ ♦ ♠❛✐♦r s❡❣♠❡♥t♦ é ♦ ❞♦❜r♦ ❞♦ ♠❡♥♦r✳. ❼. ❯♠ ♥ú♠❡r♦ ❝✉❥❛ s♦♠❛ ❞♦s ❛❧❣❛r✐s♠♦s é ♠ú❧t✐♣❧♦ ❞❡ ✾ é✱ ❡❧❡ ♣ró♣r✐♦✱ ♠ú❧t✐♣❧♦ ❞❡ ✾✳. ❙✉❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ♣r♦♣♦s✐çõ❡s ♣❛rt✐❝✉❧❛r❡s ❡♥❝♦♥tr❛♠✲s❡ ❛❜❛✐①♦✱ ♥❛ ♠❡s♠❛ ♦r❞❡♠ ❞❛s s✉❛s ❣❡r❛✐s✿. ❼. ❏♦ã♦✱ ✉♠ ♠❛r❛♥❤❡♥s❡✱ é ✐❣✉❛❧ ♣❡r❛♥t❡ ❛ ❧❡✐✱ ❛ ❋r❛♥❝✐s❝♦✱ ✉♠ ❝❡❛r❡♥s❡✳. ❼. ❖ ❜❛r✐❝❡♥tr♦. M1 ❼. G ❞♦ tr✐â♥❣✉❧♦ ABC. ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ ❧❛❞♦. ❞✐✈✐❞❡ ♦ s❡❣♠❡♥t♦. AM1. ♥❛ r❛③ã♦. AG GM1. = 2✱ s❡♥❞♦. BC ✳. ✶✵✵✵✽ é ❞✐✈✐sí✈❡❧ ♣♦r ✾✳. ❊♥tr❡t❛♥t♦✱ r❡❛❧✐③❛❞❛ ❝♦♠ ❜❛s❡ ❛♣❡♥❛s ♥❛ ✈❡r❛❝✐❞❛❞❡ ❞❡ ✭❛❧❣✉♥s✮ ❝❛s♦s ♣❛rt✐✲ ❝✉❧❛r❡s✱ ❛ ✏✐♥❞✉çã♦✑. ✶ ❆s. ✶. ♥❡♠ s❡♠♣r❡ ♥♦s ❧❡✈❛ ❛ ❝♦♥❝❧✉sõ❡s ✈❡r❞❛❞❡✐r❛s✳. ❛s♣❛s ❛í s❡r✈❡♠ ♣❛r❛ s✉❣❡r✐r q✉❡ ❛ ♣❛❧❛✈r❛. ✐♥❞✉çã♦. ❡stá s❡♥❞♦ ✉s❛❞♦ ❝♦♠ ♦ s❡♥t✐❞♦ ❞❡. ❞❡ ♣❛ss❛❣❡♠ ❞❡ ✉♠❛ ❛✜r♠❛çã♦ ♣❛rt✐❝✉❧❛r à s✉❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❣❡r❛❧✳. ♣r♦❝❡ss♦.

(16) ✶✳✶ ❘❛❝✐♦❝í♥✐♦ ✐♥❞✉t✐✈♦. ✶✸. ❱❡❥❛♠♦s ✐st♦ ♥❛s ❞✉❛s s✐t✉❛çõ❡s ❛❜❛✐①♦✳ ❙✐t✉❛çã♦ ✶. ✶✵✵✵✽ é ❞✐✈✐sí✈❡❧ ♣♦r ✾✳. ✭✶✳✶✳✶✮. ❯♠ ♥ú♠❡r♦ ❝✉❥❛ s♦♠❛ ❞♦s ❛❧❣❛r✐s♠♦s é ♠ú❧t✐♣❧♦ ❞❡ ✾ é✱ ❡❧❡ ♣ró♣r✐♦✱ ♠ú❧t✐♣❧♦ ❞❡ ✾✳ ✭✶✳✶✳✷✮ ❈♦♠ ❜❛s❡ ♥❛ ❆✜r♠❛çã♦ ✭✶✳✶✳✶✮✱ ♦❜t✐✈❡♠♦s ❛ ❆✜r♠❛çã♦ ✭✶✳✶✳✷✮✱ q✉❡ s❛❜❡♠♦s s❡r ✈❡r❞❛❞❡✐r❛✳ ❙✐t✉❛çã♦ ✷. ✶✵✵✵✽ é ❞✐✈✐sí✈❡❧ ♣♦r ✾✳. ✭✶✳✶✳✸✮. ❚♦❞♦s ♦s ♥ú♠❡r♦s q✉❡ ❝♦♠❡ç❛♠ ❝♦♠ ✶ ❡ t❡r♠✐♥❛♠ ❝♦♠ ✽ sã♦ ♠ú❧t✐♣❧♦s ❞❡ ✾✳ ✭✶✳✶✳✹✮ ❇❛s❡❛♥❞♦✲♥♦s ♥❛ Pr♦♣♦s✐çã♦ P❛rt✐❝✉❧❛r ✭✶✳✶✳✸✮✱ ❝❤❡❣❛♠♦s à Pr♦♣♦s✐çã♦ ●❡r❛❧ ✭✶✳✶✳✹✮✱ ✐♥✈❡rí❞✐❝❛✱ ❝♦♠♦ s❛❜❡♠♦s✳ ❖ ❡♠♣r❡❣♦ ❞♦ r❛❝✐♦❝í♥✐♦ ✐♥❞✉t✐✈♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❡♠ s✐t✉❛✲ çõ❡s ❡♥✈♦❧✈❡♥❞♦ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ♣❛r❛ q✉❡ r❡s✉❧t❡ ❡♠ ❝♦♥❝❧✉sã♦ ❝♦rr❡t❛✱ ❞❡✈❡✱ ❛❧é♠ ❞❡ s❡ ❜❛s❡❛r ♥❛ ✈❡r❛❝✐❞❛❞❡ ❞❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ♣❛rt✐❝✉❧❛r ✭♦✉ ❞❡ ♣r♦♣♦s✐çõ❡s ♣❛rt✐❝✉❧❛✲ r❡s✮✱ s✉❜♠❡t❡r✲s❡ ❛ ♠❛✐s ✉♠ t❡st❡✱ q✉❛❧ s❡❥❛✱ ❛ ❝♦♥st❛t❛çã♦ ❞❡ q✉❡ ❛ ✈❡r❛❝✐❞❛❞❡ ❞♦ q✉❡ s❡ ❛✜r♠❛ ♣❛r❛ ✉♠ ❞❛❞♦ ♥❛t✉r❛❧ n ❛❝❛rr❡t❛ ❛ ✈❡r❛❝✐❞❛❞❡ ❞❡st❛ ❛✜r♠❛çã♦ ♣❛r❛ ♦ ♥❛t✉r❛❧ n + 1✳ ◆✐st♦ ❝♦♥s✐st❡ ♦ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✱ ❢✉♥❞❛♠❡♥t❛❞♦ ♥♦ ♣r✐♥❝í♣✐♦✷ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✳. ▼❛s✱ ❛♥t❡s ❞❡ tr❛t❛r ❞♦ ♠ét♦❞♦ ❡ ❞♦ ♣r✐♥❝í♣✐♦✱ ✈❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ r❛❝✐♦❝í♥✐♦s ❡♠ q✉❡✱ ♣r❡t❡♥s❛♠❡♥t❡✱ s❡ ❛♣❧✐❝❛ ❛ ✐♥❞✉çã♦ ❝♦rr❡t❛♠❡♥t❡✳ ❊①❡♠♣❧♦ ✶✳✶✳✶✳. ❙❡❥❛ Sn =. 1 1 1 1 + + + ··· + , 1·3 3·5 5·7 (2n − 1)(2n + 1). ♦♥❞❡ n é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ✭q✉❛❧q✉❡r✮✳ ✷ ❆q✉✐✱. ✉t✐❧✐③♦ ❛ ♣❛❧❛✈r❛. ♣r✐♥❝í♣✐♦. ❝♦♠♦ s✐♥ô♥✐♠♦ ❞❡. ❛①✐♦♠❛✳. ❊♠❜♦r❛ ♥ã♦ ♠✉✐t♦ ❢r❡q✉❡♥t❡ ♥❛ ❧✐t❡r❛t✉r❛. s♦❜r❡ ♦ ❛ss✉♥t♦✱ ❡st❛ ♣rát✐❝❛ ♣♦❞❡ s❡r ✈✐st❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❬✼❪✱ ♣á❣✐♥❛ ✷✳.

(17) ✶✳✶. ❘❛❝✐♦❝í♥✐♦ ✐♥❞✉t✐✈♦. ❊①❛♠✐♥❛♥❞♦ ♦s ✈❛❧♦r❡s. ✶✹. Sn. ♣❛r❛ ♦s ♣r✐♠❡✐r♦s ✹ ♥❛t✉r❛✐s✱ ✈❡♠♦s q✉❡. 1 1·3 1 S2 = 1·3 1 S3 = 1·3 1 S4 = 1·3 S1 =. 1 = , 3 1 2 + = , 3·5 5 1 1 3 + + = , 3·5 5·7 7 1 1 1 4 + + + = . 3·5 5·7 7·9 9. ❈♦♠ ❜❛s❡ ♥❡st❡ ❡①❛♠❡ ♣♦❞❡♠♦s ❛✜r♠❛r✱ ❡♥tã♦✱ q✉❡✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧. n✱. t❡♠✲s❡ q✉❡. Sn = ❊①❡♠♣❧♦ ✶✳✶✳✷ ✭. ❲✐❧❤❡❧♠ ▲❡✐❜♥✐③. ▲❡✐❜♥✐③ ✮✳. ❖ ♥♦tá✈❡❧ ♠❛t❡♠át✐❝♦ ✭❥✉r✐st❛ ❡ ✜❧ós♦❢♦✮ ❛❧❡♠ã♦. ❞❡♠♦♥str♦✉ q✉❡. nk − n✱. ●♦tt❢r✐❡❞. n3 − n é ❞✐✈✐sí✈❡❧ ♣♦r ✸✱ n5 − n é ❞✐✈✐sí✈❡❧ ♣♦r ✺ ❡ n7 − n. é ❞✐✈✐sí✈❡❧ ♣♦r ✼✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❞❡. n . 2n + 1. n✳. ❆ ♣❛rt✐r ❞✐ss♦✱ ❝♦♥❝❧✉í♠♦s q✉❡. q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦ ♥ú♠❡r♦ í♠♣❛r. k. ❡ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧. k. é ❞✐✈✐s♦r. n✳. ❆♥❛❧✐s❡♠♦s✱ ❛❣♦r❛✱ ❛ r❛③ã♦ ♣❡❧❛ q✉❛❧ ♥ã♦ sã♦ ❛❝❡✐tá✈❡✐s ♦s r❛❝✐♦❝í♥✐♦s ✉t✐❧✐③❛❞♦s ♥♦s ❞♦✐s ❡①❡♠♣❧♦s ❛♥t❡r✐♦r❡s ✭❡ ❞❛í ♣♦rq✉❡ ♦ t❡r♠♦. ♣r❡t❡♥s❛♠❡♥t❡✱ ✉s❛❞♦ ❛❝✐♠❛✮✳. ❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ❝❤❡❣❛♠♦s à ✉♠❛ ✏❝♦♥❝❧✉sã♦✑ ♦✉✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❡♥✉♥❝✐❛♠♦s ✉♠❛ ♣r♦♣♦s✐çã♦ ❣❡r❛❧✱ r❡❢❡r❡♥t❡ ❛ q✉❛❧q✉❡r. n. ♥❛t✉r❛❧✱ t♦♠❛♥❞♦ ❝♦♠♦ ❜❛s❡ ♦. ❢❛t♦ ❞❡ q✉❡✱ ❡♠ ❝❛❞❛ ✉♠ ❞❡❧❡s✱ ❛ ♣r♦♣♦s✐çã♦ ♠♦str♦✉✲s❡ ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡. n✳ ❊♥tr❡t❛♥t♦✱ ❡♠❜♦r❛ ❛ ✈❡r❛❝✐❞❛❞❡ ❞❡ ✉♠❛ ❛✜r♠❛çã♦ ♣♦ss❛ s❡r ❝♦♥st❛t❛❞❛ ❡♠. ❛❧❣✉♥s ❞❡ s❡✉s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✱ ✐st♦ ♣♦r s✐ só ♥ã♦ ❣❛r❛♥t❡ ❛ ❝♦♥❝❧✉sã♦ ❞❡ s✉❛ ✈❡r❛❝✐❞❛❞❡ ✏s❡♠♣r❡✑✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦. n. ❛ss✉♠❡ t♦❞♦s ♦s ✈❛❧♦r❡s ♥❛t✉r❛✐s✳ P♦rt❛♥t♦✱ ❛✐♥❞❛ q✉❡ ❤❛❥❛. ✐♥❞í❝✐♦s ❞❡ q✉❡ s❡ tr❛t❛ ❞❡ ✉♠❛ ❛✜r♠❛çã♦ ❝♦rr❡t❛✱ é ♣r❡♠❛t✉r❛ s✉❛ ❝♦♥❝❧✉sã♦ ❝♦♠ ❜❛s❡ ❛♣❡♥❛s ❡♠ ❛❧❣✉♥s ❞❡ s❡✉s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✳ ❈❛s✉❛❧♠❡♥t❡✱ ♥♦ ❊①❡♠♣❧♦ ✶✱ ❛ ♣r♦♣♦s✐çã♦ ❣❡r❛❧ ❡♥✉♥❝✐❛❞❛ é ✈❡r❞❛❞❡✐r❛ ✭❛ ❥✉st✐✜❝❛t✐✈❛ s❡rá ❞❛❞❛ ♠❛✐s ❛❞✐❛♥t❡✮❀ ♥♦ ❡♥t❛♥t♦✱ ♥♦ ❊①❡♠♣❧♦ ✷✱ ❛ ♣r♦♣♦s✐çã♦ s❡ ♠♦str❛ ❢❛❧s❛ ♣❛r❛. n=2. ❡. k = 9✱. ♣♦r ❡①❡♠♣❧♦✱ ♣♦✐s ✾. ♥ã♦ é ❞✐✈✐s♦r. ❞❡. 29 − 2 = 510✳. ❱❡❥❛♠♦s✱ ❛ s❡❣✉✐r✱ ♠❛✐s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ♣r♦♣♦s✐çõ❡s ❝♦rr❡t❛s ❡♠ ✈ár✐♦s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✱ ♠❛s ♥ã♦ ♥♦ ❝❛s♦ ❣❡r❛❧✳. ▲❡♦♥❤❛r❞ ❊✉❧❡r ✐♥t❡r❡ss♦✉✲s❡✱ ♣❛r❛ n ✈❛r✐❛♥❞♦ ❡♥tr❡ ♦s ♥❛t✉r❛✐s ❞❡ ✶ ❛ ✸✾✱ r❡s✉❧t❛ ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳ ❙❡r✐❛ ❞❡ ❡s♣❡r❛r✱ ❞❛í✱ ❊①❡♠♣❧♦ ✶✳✶✳✸✳ ❖ tr✐♥ô♠✐♦. n2 + n + 41✱. ♣❡❧♦ q✉❛❧.

(18) ✶✳✶ ❘❛❝✐♦❝í♥✐♦ ✐♥❞✉t✐✈♦. ✶✺. q✉❡ ♣❛r❛ ♦s ❞❡♠❛✐s ♥❛t✉r❛✐s n✱ ❛ ❡①♣r❡ssã♦ n2 + n + 41 r❡s✉❧t❛ss❡✱ t❛♠❜é♠✱ ✉♠ ♣r✐♠♦✳ ◆♦ ❡♥t❛♥t♦✱ é ❢á❝✐❧ ✈❡r q✉❡ ♣❛r❛ n = 40 ❡st❡ tr✐♥ô♠✐♦ é ✐❣✉❛❧ ❛ 412 ✱ ♥ú♠❡r♦ ❝♦♠♣♦st♦✳ ❊✱ ❝❧❛r❛♠❡♥t❡✱ s❡ n = 41✱ t❡♠✲s❡ t❛♠❜é♠ ♣❛r❛ ♦ tr✐♥ô♠✐♦ ✉♠ ♥ú♠❡r♦ ❝♦♠♣♦st♦✳ ❊①❡♠♣❧♦ ✶✳✶✳✹✳. ❖ ♠❛t❡♠át✐❝♦ r✉ss♦✱ ❉✐♠✐tr✐ ●r❛✈❡ ❝♦♥❥❡❝t✉r♦✉✱ ♥♦ ✜♠ ❞♦ sé❝✉❧♦ ❳■❳✱. q✉❡✱ ♣❛r❛ t♦❞♦ ♣r✐♠♦ p✱ ✉♠ ♥ú♠❡r♦ ❞♦ t✐♣♦ 2p−1 − 1 ♥ã♦ é ❞✐✈✐sí✈❡❧ ♣♦r p2 ✳ ❈á❧❝✉❧♦s. ❞✐r❡t♦s ❞❛✈❛♠ ❝♦♥t❛ ❞❡ q✉❡ ❡st❛ ❛✜r♠❛çã♦ ❡r❛ ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦s ♦s ♥ú♠❡r♦s ♣r✐♠♦s ♠❡♥♦r❡s ❞♦ q✉❡ ✶✵✵✵✳ ▼❛s ❛ ❤✐♣ót❡s❡ ❞❡ ●r❛✈❡ r❡✈❡❧♦✉✲s❡ ❢❛❧s❛ ❝♦♠ ❛ ❞❡s❝♦❜❡rt❛✱ ♣♦r ❲✳ ▼❡✐ss♥❡r. ✱ ❡♠ ✶✾✶✸✱ ❞❡ q✉❡ ❛ ❝♦♥❣r✉ê♥❝✐❛✸ 2p−1 ≡ 1 (♠♦❞ p2 ). é s❛t✐s❢❡✐t❛✹ ♣❛r❛ ♦ ♣r✐♠♦ p = 1093✱ ♦ q✉❡ s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ 21092 − 1 é ❞✐✈✐sí✈❡❧ ♣♦r 10932 ✳ ❊①❡♠♣❧♦ ✶✳✶✳✺✳. ➱ ♣♦ssí✈❡❧ ♦❜t❡r ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ❡♥tr❡ ♦s ♥ú♠❡r♦s 991n2 + 1✱ ❛♦. s❡ ❢❛③❡r n = 1, 2, 3, . . .❄ ❆ r❡s♣♦st❛ é s✐♠✱ ❡♠❜♦r❛ ♦ ✈❛❧♦r ♠í♥✐♠♦ ❞❡ n ♣❛r❛ ♦ q✉❛❧ ♦ ♥ú♠❡r♦ 991n2 + 1 é ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ s❡❥❛ n = 12 055 735 790 331 359 447 442 538 767. ❊①❡♠♣❧♦ ✶✳✶✳✻✳. ❊♠ q✉❛♥t❛s ♣❛rt❡s n ♣❧❛♥♦s✱ ❞✐s♣♦st♦s ❞❡ t❛❧ ♠♦❞♦ q✉❡ t♦❞♦s ♣❛ss❡♠. ♣♦r ✉♠ ♠❡s♠♦ ♣♦♥t♦ ❡ ♥ã♦ ♣❛ss❡♠ três ♣♦r ✉♠❛ ♠❡s♠❛ r❡t❛✱ ❞✐✈✐❞❡♠ ♦ ❡s♣❛ç♦❄ ▲❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ q✉❡ ✉♠ ♣❧❛♥♦ ❞✐✈✐❞❡ ♦ ❡s♣❛ç♦ ❡♠ ❞✉❛s ♣❛rt❡s❀ ❞♦✐s ♣❧❛♥♦s ❝♦♠ ✉♠❛ r❡t❛ ❡♠ ❝♦♠✉♠ ✭❡✱ ♣♦rt❛♥t♦✱ ❝♦♠ ✉♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠✮ ❞✐✈✐❞❡♠ ♦ ❡s♣❛ç♦ ❡♠ q✉❛tr♦ ♣❛rt❡s❀ ❡ três ♣❧❛♥♦s✱ ❝♦♠ ❛❧❣✉♠ ♣♦♥t♦ ❡♠ ❝♦♠✉♠ ❡ ♥ã♦ ♣❛ss❛♥❞♦ t♦❞♦s ♣♦r ✉♠❛ ♠❡s♠❛ r❡t❛✱ ❞✐✈✐❞❡♠ ♦ ❡s♣❛ç♦ ❡♠ ♦✐t♦ ♣❛rt❡s✱ s♦♠♦s ❧❡✈❛❞♦s ❛ ❝♦♥❥❡❝t✉r❛r q✉❡✱ ❡♠ ❣❡r❛❧✱ n ♣❧❛♥♦s ❞✐✈✐❞❡♠ ♦ ❡s♣❛ç♦ ❡♠ 2n ♣❛rt❡s ❞✐st✐♥t❛s✳ ❊♥tr❡t❛♥t♦✱ ♣♦❞❡✲s❡ ❞❡♠♦♥str❛r✺ q✉❡ s❡ An é ♦ ♥✉♠❡r♦ ❞❡ ♣❛rt❡s ❡♠ q✉❡ n ♣❧❛♥♦s✱ ♣❛s✲ s❛♥❞♦ t♦❞♦s ♣♦r ✉♠ ♠❡s♠♦ ♣♦♥t♦ ❡ ♥ã♦ ♣❛ss❛♥❞♦ três ♣♦r ✉♠❛ ♠❡s♠❛ r❡t❛✱ ❞✐✈✐❞❡♠ ♦ ❡s♣❛ç♦✱ ❡♥tã♦ An = n(n − 1) + 2.. ❖ ♦❜❥❡t✐✈♦ ❞❛ ❡①✐❜✐çã♦ ❞♦s ❡①❡♠♣❧♦s s✉♣r❛❝✐t❛❞♦s é ♣r❡✈❡♥✐r q✉❡♠ q✉❡r q✉❡ s❡❥❛ ❞❡ ❝♦♥❝❧✉sõ❡s ❜❛s❡❛❞❛s ❡♠ ❛♥❛❧♦❣✐❛s ♠❛❧ ❛r❣✉♠❡♥t❛❞❛s✳ ✸ ❯♠ ♥ú♠❡r♦ ♣r✐♠♦. p. q✉❡ s❛t✐s❢❛③ ❛ ❝♦♥❣r✉ê♥❝✐❛. ♣r✐♠♦ ❞❡ ❲✐❡❢❡r✐❝❤. 2p−1 ≡ 1 (♠♦❞ p2 ). é ❝❤❛♠❛❞♦ ✳ ✹ ❖✉tr♦ ♣r✐♠♦ q✉❡ ❝✉♠♣r❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ❲✐❡❢❡r✐❝❤ é p = 3511 ✭❝♦♥s✉❧t❛r r❡❢❡rê♥❝✐❛ ❬✶✻❪✮✳ ✺ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣♦s✐çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥❛ r❡❢❡rê♥❝✐❛ ❬✸❪✳.

(19) ✶✳✷. ■♥❞✉çã♦ ♠❛t❡♠át✐❝❛. ✶✳✷. ✶✻. ■♥❞✉çã♦ ♠❛t❡♠át✐❝❛. ✶✳✷✳✶. ❊①❡♠♣❧♦ ❞❡ ❞❡♠♦♥str❛çã♦ ♣♦r ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛. n✱. P♦st♦ q✉❡✱ ❛♦ s❡ ❛❞✐❝✐♦♥❛r ✶ ❛♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❞♦ q✉❡. n✱. ✉♠❛ ✈❡③ q✉❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧. ✉♠ ♥❛t✉r❛❧ t♦❞♦s✳. p. t❛❧ q✉❡. n + p = m✮✱. m. ♦❜té♠✲s❡ ♦ ✐♥t❡✐r♦. n+1. é ❞✐t♦ s❡r ♠❛✐♦r ❞♦ q✉❡. n. ✭q✉❡ é ♠❛✐♦r. q✉❛♥❞♦ ❡①✐st❡. ♥ã♦ ❤á ♥ú♠❡r♦ ✐♥t❡✐r♦ ❛❧❣✉♠ q✉❡ s❡❥❛ ♦ ♠❛✐♦r ❞❡. ✻. ❆♣❡s❛r ❞✐ss♦✱ ✐♥✐❝✐❛♥❞♦ ❝♦♠ ♦ ♥ú♠❡r♦ ✶✱ t♦❞♦s ♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s. ♣♦❞❡♠ s❡r. ❛❧❝❛♥ç❛❞♦s✱ ❛♣ós ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♦♣❡r❛çõ❡s ❞❡ ✏s♦♠❛r ✶✑✱ ♣❛ss❛♥❞♦ s✉❝❡ss✐✈❛♠❡♥t❡ ❞❡. n ❛ n + 1✳. ❊st❛ é ❛ ❡ssê♥❝✐❛ ❞❡ ✉♠ ♠ét♦❞♦ ❞❡ r❛❝✐♦❝í♥✐♦ q✉❡ ♦s ♠❛t❡♠át✐❝♦s ❜❛t✐③❛r❛♠. ❞❡ ❞❡♠♦♥str❛çã♦ ♣♦r ✐♥❞✉çã♦ ♦✉ ♠ét♦❞♦ ❞❡ ✐♥❞✉çã♦ ♠❛t❡♠át✐❝❛✳. ❱❡❥❛♠♦s ❛ ❛♣❧✐❝❛çã♦. ❞❡st❡ ♠ét♦❞♦✱ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❞✉♣❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❜❛✐①♦✳. Pr♦♣♦s✐çã♦ ✶✳✷✳✶✳ ❉❛❞♦ ✉♠ ♥❛t✉r❛❧ q✉❛❧q✉❡r. 1 2 + 2 2 + · · · + n2 <. n✱. t❡♠✲s❡ q✉❡. (n + 1)3 < 12 + 22 + · · · + (n + 1)2 . 3 P (n). ❉❡♠♦♥str❛çã♦✳ ■♥✐❝✐❛❧♠❡♥t❡✱ r❡♣r❡s❡♥t❡♠♦s ♣♦r. ✭✶✳✷✳✶✮. ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❛ ❡sq✉❡r❞❛ ❞❛ ❡①✲. ♣r❡ssã♦ ❛❝✐♠❛✱ ✐st♦ é✱ ❝♦♥s✐❞❡r❡♠♦s. P (n) : 12 + 22 + · · · + n2 < ❱❡r✐✜❝❛♥❞♦ ❞✐r❡t❛♠❡♥t❡ ❛ ❛✜r♠❛çã♦ t❡r❡♠♦s q✉❡. P (1)✱ P (2). ❡. P (3). P (n). ♣❛r❛ ♦s ♣r✐♠❡✐r♦s três ✈❛❧♦r❡s ❞❡. n✱. sã♦ ❛ss❡rçõ❡s ✈❡r❞❛❞❡✐r❛s✱ ♣♦✐s. P (1) : 12 <. 23 3. P (2) : 12 + 22 <. 33 3. P (3) : 12 + 22 + 32 < ◗✉❡r❡♠♦s ♠♦str❛r q✉❡. (n + 1)3 . 3. 43 3. P (n) é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧ n ✭✐♥❞✉③✐❞♦s ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡. ❛ ❛ss❡rt✐✈❛ s❡ ✈❡r✐✜❝❛ ♣❛r❛ ♦s três ♣r✐♠❡✐r♦s ❝❛s♦s ✐♥✐❝✐❛✐s✮✳ ■r❡♠♦s s✉♣♦r q✉❡ ❛ ♣r♦♣♦s✐çã♦ s❡ ✈❡r✐✜❝❛ ♣❛r❛ ✉♠ ❞❛❞♦ ✈❛❧♦r ♣❛rt✐❝✉❧❛r. n = k > 1✳. ■st♦ q✉❡r ❞✐③❡r q✉❡ ❝♦♥s✐❞❡r❛r❡♠♦s. ♣r♦✈❛❞❛ ❛ ❛✜r♠❛çã♦. P (k) : 12 + 22 + · · · + k 2 < ✻ ❆q✉✐✱. (k + 1)3 , 3. s❡rã♦ ❝♦♥s✐❞❡r❛❞♦s ❝♦♠♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦. N = {1, 2, 3, . . .}✳.