❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❚r✐â♥❣✉❧♦ ▼✐♥❡✐r♦
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚
●✉st❛✈♦ ❆❧✈❡s ❈❛❡t❛♥♦ ◆❡t♦
❯▼❆ ■❉❊■❆ ❙❖❇❘❊ ❖ ❈❖◆❈❊■❚❖ ❉❊ ▲■▼■❚❊ ❆❖ ▲❖◆●❖ ❉❆ ❍■❙❚Ó❘■❆ ❉❆ ▼❆❚❊▼➪❚■❈❆
●✉st❛✈♦ ❆❧✈❡s ❈❛❡t❛♥♦ ◆❡t♦
❯▼❆ ■❉❊■❆ ❙❖❇❘❊ ❖ ❈❖◆❈❊■❚❖ ❉❊ ▲■▼■❚❊ ❆❖ ▲❖◆●❖ ❉❆ ❍■❙❚Ó❘■❆ ❉❆ ▼❆❚❊▼➪❚■❈❆
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❝✉rs♦ ❞❡ ▼❡s✲ tr❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚✱ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡✲ ❞❡r❛❧ ❞♦ ❚r✐â♥❣✉❧♦ ▼✐♥❡✐r♦✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❛t✐✈✐❞❛❞❡s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❛✳ ❉r❛✳ ▼ô♥✐❝❛ ❞❡ ❈áss✐❛ ❙✐q✉❡✐r❛ ▼❛rt✐♥❡s
C a t a l o g a ç ã o n a f o n t e : B i b l i o t e c a d a U n i v e r s i d a d e F e d e r a l d o T r i â n g u l o M i n e i r o
Caetano Neto, Gustavo Alves
C131i Uma ideia sobre o conceito de limite ao longo da história da matemática / Gustavo Alves Caetano Neto. -- 2016.
118 f. : il.
Dissertação (Mestrado Profissional em Matemática em Rede Nacional) -- Universidade Federal do Triângulo Mineiro, Uberaba, MG, 2016
Orientadora: Profª Drª Mônica de Cássia Siqueira Martines
1. Matemática - História. 2. Cálculo. I. Martines, Mônica de Cássia ra. II. Universidade Federal do Triângulo Mineiro. III. Título.
✏❚✉❞♦ ♦ q✉❡ ✉♠ s♦♥❤♦ ♣r❡❝✐s❛ ♣❛r❛ s❡r r❡❛❧✐③❛❞♦ é ❛❧❣✉é♠ q✉❡ ❛❝r❡❞✐t❡ q✉❡ ❡❧❡ ♣♦ss❛ s❡r r❡❛❧✐③❛❞♦✑✳
❆●❘❆❉❊❈■▼❊◆❚❖❙
❆❣r❛❞❡ç♦✱ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ▼ô♥✐❝❛✱ ♣❡❧♦ s❡✉ ❥❡✐t♦ ú♥✐❝♦ ❞❡ ♦r✐❡♥t❛r✱ ♣❡❧❛ s✉❛ ❣r❛♥❞❡ ❤✉♠✐❧❞❛❞❡ ❡ ♣❡❧♦ ❝❛r✐♥❤♦ ❞❡♠♦♥str❛❞♦ ♣♦r ♠✐♠✱ s❡♠♣r❡ ❡❧❡✈❛♥❞♦ ❛ ♠✐♥❤❛ ❛✉t♦✲❡st✐♠❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦s ♠♦♠❡♥t♦s ❞❡ ✐♥❝❡rt❡③❛✱ ♠❡ ❢❛③❡♥❞♦ ❛❝r❡❞✐t❛r q✉❡ s♦✉ ❡s♣❡❝✐❛❧ ❡ q✉❡ ♣♦ss♦ ❝♦♥q✉✐st❛r ♠❛✐s ❛ ❝❛❞❛ ❞✐❛✱ q✉❡ t✉❞♦ ❞❡♣❡♥❞❡ ❞♦ ♠❡✉ ❡s❢♦rç♦✳ P❡♥s♦ t❡r t✐❞♦ ❡♠ ▼ô♥✐❝❛ ✉♠ ❜❡❧♦ ❡①❡♠♣❧♦ ❞❡ q✉❡ ♠❡s♠♦ ✉♠ ♣r♦❢❡ss♦r ✏❉♦✉t♦r✑✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ s❡✉ ♥í✈❡❧ ✐♥t❡❧❡❝t✉❛❧ ♦✉ ❞❡ s✉❛ ♣♦s✐çã♦✱ ♣♦❞❡ ❝♦♥t✐♥✉❛r s❡♥❞♦ ✉♠❛ ♣❡ss♦❛ s✐♠♣❧❡s✱ ♣❛❝✐❡♥t❡✱ ❤✉♠✐❧❞❡✱ s❡❞❡♥t❛ ♣♦r ❡♥s✐♥❛r ❜❡♠ ♦s s❡✉s ❛❧✉♥♦s ❡✱ t❛♠❜é♠✱ s❡❞❡♥t❛ ♣♦r ❛♣r❡♥❞❡r ❝♦♠ ❡❧❡s✳
❆♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s q✉❡ ♠♦r❛♠ ❡♠ ♦✉tr❛ ❝✐❞❛❞❡✱ ♣❡❧❛s ♦r❛çõ❡s ❡♠ ♠❡✉ ❢❛✈♦r✱ t♦r❝❡♥❞♦ s❡♠♣r❡ ♣❡❧♦ ♠❡✉ s✉❝❡ss♦✳
❆♦s ♠❡✉s s❛✉❞♦s♦s ♣❛❞r✐♥❤♦s✱ ❆✐rt♦♥ ❡ ❏✉✈❡❧✐♥❛ ✭❚✐t❛✮✱ q✉❡ ❛❝♦❧❤❡r❛♠ ❛ ♠✐♠ ❡ à ♠✐♥❤❛ ♠ã❡ ❡♠ s✉❛ ❝❛s❛ ❡ ❝♦♥tr✐❜✉ír❛♠ ♠✉✐t♦ ♣❛r❛ ❛ ♠✐♥❤❛ ❡❞✉❝❛✲ çã♦✱ ❛❧é♠ ❞❡ s❡r❡♠ ❣r❛♥❞❡s ✐♥❝❡♥t✐✈❛❞♦r❡s ❞❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦s ♠❡✉s ❡st✉❞♦s✳ ➚ ♠✐♥❤❛ ♥❛♠♦r❛❞❛✱ ■✈❛♥✐❧❞❛✱ ♣❡❧♦ ❛♠♦r ❡ ❝❛r✐♥❤♦ ✐♥❝♦♠♣❛rá✈❡✐s ❡ ♣❡❧♦ ❛♣♦✐♦ ♥❛s ❤♦r❛s ❞❡ ❞✐✜❝✉❧❞❛❞❡✳
❆♦s ❛♠✐❣♦s q✉❡ ♣❛ss❛r❛♠ ♣♦r ♠✐♥❤❛ ✈✐❞❛✱ ❞❡s❞❡ ❛ ✐♥❢â♥❝✐❛ ❛té ❛❣♦r❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❆✉ré❧✐♦✱ ❏❡✛❡rs♦♥✱ ❆♠❛♥❞❛✱ ▼❛✉rí❝✐♦✱ P❛trí❝✐❛✱ ❆♥❞ré✱ ❈❛r❧♦s ❍❡♥r✐q✉❡✱ ❋r❡❞❡r✐❝♦ ❡ ❘♦❜s♦♥✱ q✉❡ ♠❡ ❞❡r❛♠ ♠✉✐t❛ ❢♦rç❛ ❡♠ ♠✉✐t♦s ♠♦♠❡♥t♦s ❞❡ ♠✐♥❤❛ tr❛❥❡tór✐❛✳
➚s ❢r❡✐r❛s ❞♦ ❈♦❧é❣✐♦ ◆♦ss❛ ❙❡♥❤♦r❛ ❞❛s ❉♦r❡s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ às ■r♠ãs ◆❛❞✐r ❡ ▼❛r✐❛ ❍❡❧❡♥❛✱ ♣♦r ♠❡ ❝♦♥❝❡❞❡r❡♠✱ ❞✉r❛♥t❡ t♦❞♦ ♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥✲ t❛❧✱ ❜♦❧s❛ ✐♥t❡❣r❛❧ ❞❡ ❡st✉❞♦s ❡♠ s✉❛ ❝♦♥❝❡✐t✉❛❞❛ ❡s❝♦❧❛ ❡ ♣♦r ♠❡ tr❛♥s♠✐t✐r❡♠ s❡♠♣r❡ ✈❛❧♦r❡s ❝♦♠♦ ❢é✱ r❡s♣❡✐t♦✱ ❞✐s❝✐♣❧✐♥❛✱ ❝♦♠♣❛✐①ã♦ ❡ ❤✉♠✐❧❞❛❞❡✱ s❡♥❞♦✱ ♣♦rt❛♥t♦✱ ♣❡ss♦❛s ❡ss❡♥❝✐❛✐s ❡♠ ♠✐♥❤❛ ❢♦r♠❛çã♦✳
❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ♠é❞✐♦✱ q✉❡ ♠❡ ❞❡r❛♠ ❛ ❜❛s❡ ♥❡❝❡ssár✐❛ ♣❛r❛ ❝♦♥t✐♥✉❛r ♦s ♠❡✉s ❡st✉❞♦s✳
♥❛♥❞♦✱ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ❡❞✉❝❛çã♦ q✉❡ s❡♠♣r❡ t✐✈❡r❛♠ ❝♦♠✐❣♦ ❡ ♣♦r ♠❡ ❧❡✈❛r ❛ ❜✉s❝❛r ♠❛✐s ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ❛ s❡r ✉♠❛ ♣❡ss♦❛ ♠❛✐s ❞✐s❝✐♣❧✐♥❛❞❛✳
❆♦s ❛♠✐❣♦s q✉❡ ✜③ ♥❛ ❣r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ♣❡❧❛ ❯❋❚▼ ❡ ♥♦ P❘❖❋▼❆❚✱ ♣❡❧❛s ✐♥❝♦♥tá✈❡✐s ❤♦r❛s ❞❡ r✐s♦s ❢á❝❡✐s✱ s❡♥❞♦ ❛ ♠♦t✐✈❛çã♦ ♥❡✲ ❝❡ssár✐❛ ♣❛r❛ q✉❡ ❡✉ ❝♦♥❝❧✉íss❡ ♦s ♠❡✉s ❝✉rs♦s✳
❆♦s ❛♠✐❣♦s q✉❡ ✜③ ♥❛ ❊s❝♦❧❛ ❊st❛❞✉❛❧ ▲❛✉r♦ ❋♦♥t♦✉r❛✱ ❞❡s❞❡ q✉❡ ❝♦♠❡✲ ❝❡✐ ❛ ❧❡❝✐♦♥❛r ♣❛r❛ ♦ ❡♥s✐♥♦ ❜ás✐❝♦ ❡♠ ♠❛✐♦ ❞❡ ✷✵✶✸✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛♦s ♣r♦❢❡s✲ s♦r❡s ■s❛í❛s✱ P❛✉❧♦ ❘♦❜❡rt♦✱ ❈r✐st✐♥❛✱ ❲❡s❧❡✐✱ ▼❛r✐â♥❣❡❧❛✱ ❙ér❣✐♦ ❡ ❚❤❛♠✐r✐s ❡ ❛♦s ❢✉♥❝✐♦♥ár✐♦s ▼ár❝✐♦✱ ▼❛r✐❛ ❉❡♥❡✐❞❡ ❡ ❍❡❧♦ís❛✱ ♣❡❧❛ tr♦❝❛ ❞❡ ❡①♣❡r✐ê♥❝✐❛s q✉❡ ♠✉✐t♦ ♠❡ ❛❥✉❞♦✉ ♥❡st❡ ✐♥í❝✐♦ ❞❡ ❝❛rr❡✐r❛ ❝♦♠♦ ♣r♦❢❡ss♦r✳
➚ ❈❆P❊❙✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✱ q✉❡ ♠❡ ❞❡✉ tr❛♥q✉✐❧✐❞❛❞❡ ❡ t❡♠♣♦ ♣❛r❛ ❝✉rs❛r ❛ ❣r❛❞✉❛çã♦ ❡ ♦ ♠❡str❛❞♦ ❞❛ ♠❛♥❡✐r❛ ❝♦♠♦ ❣♦st❛r✐❛✳
❘❊❙❯▼❖
❊st❡ tr❛❜❛❧❤♦ ❜✉s❝❛ ❡♥t❡♥❞❡r ❝♦♠♦ s❡ ❞❡s❡♥✈♦❧✈❡✉ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ ❛♦ ❧♦♥❣♦ ❞❡ ♣❛rt❡ ❞❛ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✱ ❞❡s❞❡ s✉❛s ❢♦r♠❛s ♠❛✐s r✉❞✐♠❡♥t❛✲ r❡s✱ ♣❛ss❛♥❞♦ ♣♦r ❛❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛✱ tr❛③❡♥❞♦ t❛♠✲ ❜é♠ ❛❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❛ ▼❛t❡♠át✐❝❛ ❞❡s❡♥✈♦❧✈✐❞❛ ♥❛ ❊✉r♦♣❛ ▼❡❞✐❡✈❛❧ ❡✱ ❞❡♣♦✐s✱ ❛❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❡ ❇♦♥❛✈❡♥t✉r❛ ❈❛✈❛❧✐❡r✐ ✭✶✺✾✽✲✶✻✹✼✮✳ Pr♦✲ ❝✉r❛♠♦s t❛♠❜é♠ ❝♦♥❤❡❝❡r ❛s ❞✐✜❝✉❧❞❛❞❡s ❡♥❢r❡♥t❛❞❛s ♣❡❧♦s ♠❛t❡♠át✐❝♦s ❞❡st❡ ♣❡rí♦❞♦ ♣❛r❛ ❡♥t❡♥❞❡r ♦s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❧✐♠✐t❡✱ t❛✐s ❝♦♠♦✿ ✏✐♥✜✲ ♥✐t❛♠❡♥t❡ ♣❡q✉❡♥♦✑✱ ✏✐♥✜♥✐t❛♠❡♥t❡ ❣r❛♥❞❡✑✱ ✐♥❞✐✈✐sí✈❡✐s✱ s♦♠❛ ❞❡ s❡q✉ê♥❝✐❛s ✐♥✜♥✐t❛s✱ ❡t❝✳ ❊♠ r❡❧❛çã♦ ❛ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ ❧✐♠✐t❡ ♣r❡s❡♥t❡s ♥❛ ▼❛t❡♠á✲ t✐❝❛ ●r❡❣❛✱ ❞❡st❛❝❛♠♦s ❛❧❣✉♥s ♣❛r❛❞♦①♦s ❞❡ ❩❡♥ã♦ ❞❡ ❊❧❡✐❛ ✭≈ ✹✻✵ ❛✳❡✳❈✳✮✱ ❛ ✐❞❡✐❛ ❞♦s ✐♥❞✐✈✐sí✈❡✐s ❡ ❛❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞♦ ❧✐✈r♦ ✏❖s ❊❧❡♠❡♥t♦s✑✳ P♦s✲ t❡r✐♦r♠❡♥t❡✱ ❛✐♥❞❛ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛✱ ❢♦❝❛♠♦s ❡♠ ❛❧❣✉♥s tr❛❜❛❧❤♦s ❞❡ ❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛ ✭≈ ✷✽✼✲✷✶✷ ❛✳❡✳❈✳✮ s♦❜r❡ ❛ q✉❛❞r❛t✉r❛ ❞♦ ❝ír❝✉❧♦✱ ❛ q✉❛❞r❛t✉r❛ ❞❛ ♣❛rá❜♦❧❛ ❡ s♦❜r❡ ❛s ❡s♣✐r❛✐s✳ ❊♠ s❡❣✉✐❞❛✱ tr❛✲ ③❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ ❧✐♠✐t❡ ♣r❡s❡♥t❡s ♥❛ ❊✉r♦♣❛ ▼❡❞✐❡✈❛❧✱ ❞❡st❛❝❛♥❞♦ ❛❧❣✉♠❛s ❞✐s❝✉ssõ❡s s♦❜r❡ ♦ ✐♥✜♥✐t♦✳ ❋✐♥❛❧✐③❛♠♦s ❡st❡ tr❛❜❛❧❤♦ ❝♦♠ ❛❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ♦ ❈á❧❝✉❧♦ ■♥t❡❣r❛❧✳ P♦❞❡♠♦s ♣❡r❝❡❜❡r✱ ♠❡s♠♦ q✉❡ ✐♠♣❧✐❝✐t❛♠❡♥t❡✱ ✉♠ ❝♦♥❝❡✐t♦ ♣r✐♠✐t✐✈♦ ❞❡ ❧✐♠✐t❡ ♣r❡s❡♥t❡ ❡♠ ✈ár✐♦s ❞♦s ♠♦♠❡♥t♦s ❛♣r❡s❡♥t❛❞♦s✱ t❛✐s ❝♦♠♦✿ ❡♠ ❛♣r♦①✐♠❛çõ❡s ♣❛r❛ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦✱ ♥❛ ✉t✐❧✐③❛çã♦ ❞❡ s♦♠❛s ✐♥✜♥✐t❛s✱ ♥❛ ❞✐s❝✉ssã♦ s♦❜r❡ ♦s ✐♥❞✐✈✐sí✈❡✐s✱ ♥❛ ✉t✐❧✐③❛✲ çã♦ ❞♦ ❧✐♠✐t❡ ❞❛s ár❡❛s ❞❡ ✜❣✉r❛s ❝✐r❝✉♥s❝r✐t❛s ❡✴♦✉ ✐♥s❝r✐t❛s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✜❣✉r❛s ❝✉r✈❛s✳
❆❇❙❚❘❆❈❚
❚❤✐s ✇♦r❦ s❡❡❦s t♦ ✉♥❞❡rst❛♥❞ ❤♦✇ ✐t ❞❡✈❡❧♦♣❡❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❧✐♠✐t ❛❧♦♥❣ ♣❛rt ♦❢ t❤❡ ❤✐st♦r② ♦❢ ♠❛t❤❡♠❛t✐❝s✱ ❢r♦♠ ✐ts ♠♦st r✉❞✐♠❡♥t❛r② ❢♦r♠s✱ ❣♦✐♥❣ t❤r♦✉❣❤ s♦♠❡ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ ●r❡❡❦ ▼❛t❤❡♠❛t✐❝s✱ ❛❧s♦ ❜r✐♥❣✐♥❣ s♦♠❡ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ ♠❛t❤❡♠❛t✐❝s ❞❡✈❡❧♦♣❡❞ ✐♥ ▼❡❞✐❡✈❛❧ ❊✉r♦♣❡ ❛♥❞ t❤❡♥ s♦♠❡ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ ❇♦♥❛✈❡♥t✉r❛ ❈❛✈❛❧✐❡r✐ ✭✶✺✾✽✲✶✻✹✼✮✳ ❲❡ ❛❧s♦ s❡❡❦ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❞✐✣❝✉❧t✐❡s ❢❛❝❡❞ ❜② ♠❛t❤❡♠❛t✐❝✐❛♥s ♦❢ t❤✐s ♣❡r✐♦❞ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❝♦♥❝❡♣ts r❡❧❛t❡❞ t♦ ❧✐♠✐t✱ s✉❝❤ ❛s✿ ✏✐♥✜♥✐t❡❧② s♠❛❧❧✑✱ ✏✐♥✜♥✐t❡❧② ❧❛r❣❡✑✱ ✐♥❞✐✈✐s✐❜❧❡s✱ s✉♠ ♦❢ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡s✱ ❡t❝✳ ■♥ r❡❧❛t✐♦♥ t♦ s♦♠❡ ❝♦♥❝❡♣ts ♦❢ ❧✐♠✐t ♣r❡s❡♥t ✐♥ t❤❡ ●r❡❡❦ ▼❛t❤❡♠❛t✐❝s✱ ✇❡ ❤✐❣❤❧✐❣❤t s♦♠❡ ♣❛r❛❞♦①❡s ♦❢ ❩❡♥♦ ♦❢ ❊❧❡❛ ✭≈✹✻✵ ❜✳❈✳❡✳✮✱ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ✐♥❞✐✈✐s✐❜❧❡s ❛♥❞ s♦♠❡ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ t❤❡ ❜♦♦❦ ✏❚❤❡ ❊❧❡♠❡♥ts✑✳ ▲❛t❡r✱ st✐❧❧ r❡❧❛t❡❞ t♦ t❤❡ ●r❡❡❦ ▼❛t❤❡♠❛t✐❝s✱ ✇❡ ❢♦❝✉s ♦♥ s♦♠❡ ✇♦r❦s ♦❢ t❤❡ ❆r❝❤✐♠❡❞❡s ♦❢ ❙②r❛❝✉s❡ ✭≈✷✽✼✲✷✶✷ ❜✳❈✳❡✳✮ ❛❜♦✉t t❤❡ q✉❛❞r❛t✉r❡ ♦❢ t❤❡ ❝✐r❝❧❡✱ t❤❡ q✉❛❞r❛t✉r❡ ♦❢ t❤❡ ♣❛r❛❜♦❧❛ ❛♥❞ ❛❜♦✉t t❤❡ s♣✐r❛❧s✳ ❚❤❡♥✱ ✇❡ ❜r✐♥❣ s♦♠❡ ❝♦♥❝❡♣ts ♦❢ ❧✐♠✐t ♣r❡s❡♥t ✐♥ t❤❡ ▼❡❞✐❡✈❛❧ ❊✉r♦♣❡✱ ❤✐❣❤❧✐❣❤t✐♥❣ s♦♠❡ ❞✐s❝✉ss✐♦♥s ❛❜♦✉t ✐♥✜♥✐t②✳ ❲❡ ❡♥❞ t❤✐s ✇♦r❦ ✇✐t❤ s♦♠❡ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ ❈❛✈❛❧✐❡r✐ t♦ t❤❡ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s✳ ❲❡ ❝❛♥ s❡❡✱ ❡✈❡♥ ✐❢ ✐♠♣❧✐❝✐t❧②✱ ❛ ♣r✐♠✐t✐✈❡ ❝♦♥❝❡♣t ♦❢ ❧✐♠✐t ✐♥ s❡✈❡r❛❧ ♦❢ t❤❡ ♣r❡s❡♥t❡❞ ♠♦♠❡♥ts✱ s✉❝❤ ❛s✿ ✐♥ ❛♣♣r♦①✐♠❛t✐♦♥s t♦ ❛r❡❛ ♦❢ t❤❡ ❝✐r❝❧❡✱ ✐♥ t❤❡ ✉s❡ ♦❢ ✐♥✜♥✐t❡ s✉♠s✱ ✐♥ t❤❡ ❞✐s❝✉ss✐♦♥ ❛❜♦✉t t❤❡ ✐♥❞✐✈✐s✐❜❧❡s✱ ✐♥ t❤❡ ✉s❡ ♦❢ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❛r❡❛s ♦❢ ✜❣✉r❡s ❝✐r❝✉♠s❝r✐❜❡❞ ❛♥❞✴♦r ❡♥r♦❧❧❡❞ ❢♦r t❤❡ ❝❛❧❝✉❧❛t✐♥❣ ♦❢ t❤❡ ❛r❡❛ ♦❢ ❝✉r✈❡s ✜❣✉r❡s✳
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙
✶✳✶ ❚r❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ❡❧❡♠❡♥t❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✷ ◗✉❛❞r❛❞♦s ✐♥s❝r✐t♦s ❡ ❝✐r❝✉♥s❝r✐t♦s ❛♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ r ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✸ ◗✉❛❞r❛♥t❡ ❝✐r❝✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✹ ❏♦♥❡s ✭✶✼✵✻✮✱ ♠♦str❛♥❞♦ ♦ sí♠❜♦❧♦ π✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✺ ❘❡♣r❡s❡♥t❛çã♦ ❞♦s ♥ú♠❡r♦s tr✐❛♥❣✉❧❛r❡s✱ q✉❛❞r❛♥❣✉❧❛r❡s✱ ❡ r❡✲
t❛♥❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✻ ❘❡♣r❡s❡♥t❛çã♦ ❞❛s ♣✐râ♠✐❞❡s tr✐❛♥❣✉❧❛r❡s✱ q✉❛❞r❛♥❣✉❧❛r❡s ❡ ❞♦s
❙❯▼➪❘■❖
■◆❚❘❖❉❯➬➹❖ ✶✹
❈❆P❮❚❯▲❖ ✶✿ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ❞❡ ❧✐♠✐t❡ ❡♠ ❝✐✈✐❧✐③❛✲
çõ❡s ❛♥t❡s ❞♦s ●r❡❣♦s ✶✾
❈❆P❮❚❯▲❖ ✷✿ ❯♠ r❡❧❛t♦ s♦❜r❡ ❛ q✉❡stã♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ♥❛
▼❛t❡♠át✐❝❛ ●r❡❣❛ ✷✺
✷✳✶ ❆ ❊s❝♦❧❛ P✐t❛❣ór✐❝❛ ❡ ♦s ♥ú♠❡r♦s ✜❣✉r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ ❖s ✐♥❞✐✈✐sí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✸ ❆❧❣✉♥s ♣❛r❛❞♦①♦s ❞❡ ❩❡♥ã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✹ ❖ ♣r♦❜❧❡♠❛ ❞♦s ✐♥❝♦♠❡♥s✉rá✈❡✐s ❡ ♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✿ ♦✉tr❛
q✉❡stã♦ ❞♦ ✐♥✜♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✺ ❆❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❡ ✏❖s ❊❧❡♠❡♥t♦s✑ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ❈❆P❮❚❯▲❖ ✸✿ ❆❧❣✉♥s tr❛❜❛❧❤♦s ❞❡ ❆rq✉✐♠❡❞❡s ❡ ✉♠ ♣♦✉❝♦ ❞❛
▼❛t❡♠át✐❝❛ ●r❡❣❛ ❛♣ós ❊✉❝❧✐❞❡s ✹✻
❈❆P❮❚❯▲❖ ✹✿ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s s♦❜r❡ ❧✐♠✐t❡ ♥❛ ❊✉r♦♣❛ ▼❡❞✐❡✈❛❧
❡ ♥♦ sé❝✉❧♦ ❳❱■■ ✾✺
✹✳✶ ❆❧❣✉♠❛s ❞✐s❝✉ssõ❡s s♦❜r❡ ♦ ✐♥✜♥✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✹✳✷ ❆❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ♦ ❈á❧❝✉❧♦ ■♥t❡❣r❛❧ ✳ ✳ ✳ ✾✽
❈❖◆❙■❉❊❘❆➬Õ❊❙ ❋■◆❆■❙ ✶✶✹
■◆❚❘❖❉❯➬➹❖ ✶✹
■◆❚❘❖❉❯➬➹❖
◆♦ ❡♥s✐♥♦ s✉♣❡r✐♦r✱ ♥❛s ❞✐s❝✐♣❧✐♥❛s ❞❡ ✏❈á❧❝✉❧♦✑✱ ❣❡r❛❧♠❡♥t❡ é ❡♥s✐♥❛❞♦ ♣r✐♠❡✐r❛♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡✱ q✉❡ ♥♦rt❡✐❛ t♦❞♦s ♦s ♦✉tr♦s ❝♦♥❝❡✐t♦s ❢✉♥✲ ❞❛♠❡♥t❛✐s ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ t❛✐s ❝♦♠♦ ❞❡r✐✈❛❞❛s✱ ✐♥t❡❣r❛✐s✱ ❡t❝✳ P♦ré♠✱ ♥❛ ❤✐stór✐❛ ❞♦ ❈á❧❝✉❧♦✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ só ❢♦✐ ❜❡♠ ❢✉♥❞❛♠❡♥✲ t❛❞♦ ❡♠ ♠❡❛❞♦s ❞♦ sé❝✉❧♦ ❳■❳✱ ♠✉✐t♦ t❡♠♣♦ ❛♣ós ❛ ✏✐♥✈❡♥çã♦✑ ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✳
❆s ❞✐s❝✐♣❧✐♥❛s ❞❡ ❈á❧❝✉❧♦ tê♠ t✐❞♦ ❛❧t♦s í♥❞✐❝❡s ❞❡ r❡♣r♦✈❛çã♦ ♥♦s ❝✉r✲ s♦s ✭❊♥❣❡♥❤❛r✐❛s✱ ▼❛t❡♠át✐❝❛✱ ❋ís✐❝❛✱ ◗✉í♠✐❝❛✱ ❡t❝✳✮ ♦♥❞❡ sã♦ ♠✐♥✐str❛❞❛s✳ ❇❛r✉✜ ✭✶✾✾✾✮ ♥♦s ♠♦str❛ ♦s í♥❞✐❝❡s ❞❡ r❡♣r♦✈❛çã♦ ♥♦s ❝✉rs♦s ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡✲ r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ❞❛ ❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❛ ❯❙P✱ ❡♥tr❡ ✶✾✾✵ ❡ ✶✾✾✺✱ q✉❡ ✈❛r✐❛♠ ❞❡20%❛75%✱ ♥❛s ❞✐s❝✐♣❧✐♥❛s ❡♠ ❣❡r❛❧ ❡ ♥ã♦ sã♦ ✐♥❢❡r✐♦r❡s ❛40%✱ ♥♦ ■♥st✐t✉t♦
❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✳ ❘❡③❡♥❞❡ ✭✷✵✵✸✮ ✈❡r✐✜❝❛ q✉❡ ♦s í♥❞✐❝❡s ❞❡ r❡♣r♦✲ ✈❛çã♦ ♥❡st❛s ❞✐s❝✐♣❧✐♥❛s✱ ♥❛ ❢❛❝✉❧❞❛❞❡ ♦♥❞❡ ❧❡❝✐♦♥❛ ✭❯❋❋✮✱ ♥♦ ♣❡rí♦❞♦ ❞❡ ✶✾✾✻ ❛ ✷✵✵✵✱ sã♦ ♠❛✐♦r❡s ❛✐♥❞❛✱ ✈❛r✐❛♥❞♦ ❞❡45% ❛ 95%✱ ♥❛s ❞✐s❝✐♣❧✐♥❛s ❡♠ ❣❡r❛❧ ❡
♥ã♦ sã♦ ✐♥❢❡r✐♦r❡s ❛65%✱ ♥♦ ❈✉rs♦ ❞❡ ▼❛t❡♠át✐❝❛✳
◆❛s❝✐♠❡♥t♦ ✭✷✵✵✸✮✱ ❡♠ s✉❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦✱ ✐♥✈❡st✐❣♦✉ ♦ ❝❡♥ár✐♦ ❞❡ r❡♣r♦✈❛çõ❡s ✈✐✈❡♥❝✐❛❞♦ ♣♦r ❡❧❡ ♥❛s ❞✐s❝✐♣❧✐♥❛s ❞❡ ❈á❧❝✉❧♦✱ ♣r✐♠❡✐r♦ ❝♦♠♦ ❛❧✉♥♦ ❡ ❞❡♣♦✐s ❝♦♠♦ ❞♦❝❡♥t❡ ❞❡st❛s ❞✐s❝✐♣❧✐♥❛s✱ ♥♦ ❡st❛❞♦ ❞❛ ❇❛❤✐❛✳ ❖ r❡❢❡r✐❞♦ ❛✉t♦r ❝♦♥s✐❞❡r❛ q✉❡ ❛s r❡♣r♦✈❛çõ❡s ♥ã♦ ❡stã♦ ❛ss♦❝✐❛❞❛s ❛♣❡♥❛s à ♦r❣❛♥✐③❛çã♦ ❡ ❡❧❛❜♦r❛çã♦ ❞♦s ❝♦♥t❡ú❞♦s✱ ❛ ✉♠ ♣r♦❣r❛♠❛ ❞❡ ❞✐s❝✐♣❧✐♥❛ s♦❜r❡❝❛rr❡❣❛❞♦✱ à ❡str✉t✉r❛ ❞❛s ❛✈❛❧✐❛çõ❡s ♦✉ ❛ ❢❛❧t❛ ❞❡ ❜❛s❡ ❞♦s ❡st✉❞❛♥t❡s ❡ s✐♠✱ ❛♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠ ❞❡st❛s ❞✐s❝✐♣❧✐♥❛s✳
■◆❚❘❖❉❯➬➹❖ ✶✺
❡st✉❞❛♥t❡s é ❛ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳ ▼❛s ♣❛r❛ q✉❡ s❡r✈❡ ❛ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛❄
❉✬❆♠❜r♦s✐♦ ✭✷✵✵✵✮✱ ❡♠ s❡✉ ❛rt✐❣♦ ✏❆ ✐♥t❡r❢❛❝❡ ❡♥tr❡ ❍✐stór✐❛ ❡ ▼❛✲ t❡♠át✐❝❛✿ ✉♠❛ ✈✐sã♦ ❤✐stór✐❝♦✲♣❡❞❛❣ó❣✐❝❛✑✱ ❡❧❡♥❝❛ q✉❛tr♦ ✜♥❛❧✐❞❛❞❡s ♣❛r❛ ❛ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✿
• ♣❛r❛ s✐t✉❛r ❛ ▼❛t❡♠át✐❝❛ ❝♦♠♦ ✉♠❛ ♠❛♥✐❢❡st❛çã♦ ❝✉❧✲
t✉r❛❧ ❞❡ t♦❞♦s ♦s ♣♦✈♦s ❡♠ t♦❞♦s ♦s t❡♠♣♦s✱ ❝♦♠♦ ❛ ❧✐♥❣✉❛❣❡♠✱ ♦s ❝♦st✉♠❡s✱ ♦s ✈❛❧♦r❡s✱ ❛s ❝r❡♥ç❛s ❡ ♦s ❤á✲ ❜✐t♦s✱ ❡ ❝♦♠♦ t❛❧ ❞✐✈❡rs✐✜❝❛❞❛ ♥❛s s✉❛s ♦r✐❣❡♥s ❡ ♥❛ s✉❛ ❡✈♦❧✉çã♦❀
• ♣❛r❛ ♠♦str❛r q✉❡ ❛ ▼❛t❡♠át✐❝❛ q✉❡ s❡ ❡st✉❞❛ ♥❛s ❡s✲
❝♦❧❛s é ✉♠❛ ❞❛s ♠✉✐t❛s ❢♦r♠❛s ❞❡ ▼❛t❡♠át✐❝❛ ❞❡s❡♥✲ ✈♦❧✈✐❞❛s ♣❡❧❛ ❤✉♠❛♥✐❞❛❞❡❀
• ♣❛r❛ ❞❡st❛❝❛r q✉❡ ❡ss❛ ▼❛t❡♠át✐❝❛ t❡✈❡ s✉❛ ♦r✐❣❡♠ ♥❛s
❝✉❧t✉r❛s ❞❛ ❛♥t✐❣✉✐❞❛❞❡ ♠❡❞✐t❡rrâ♥❡❛ ❡ s❡ ❞❡s❡♥✈♦❧✈❡✉ ❛♦ ❧♦♥❣♦ ❞❛ ■❞❛❞❡ ▼é❞✐❛ ❡ s♦♠❡♥t❡ ❛ ♣❛rt✐r ❞♦ sé❝✉❧♦ ❳❱■■ s❡ ♦r❣❛♥✐③♦✉ ❝♦♠♦ ✉♠ ❝♦r♣♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s✱ ❝♦♠ ✉♠ ❡st✐❧♦ ♣ró♣r✐♦❀
• ♣❛r❛ s❛❜❡r q✉❡ ❞❡s❞❡ ❡♥tã♦ ❛ ▼❛t❡♠át✐❝❛ ❢♦✐ ✐♥❝♦r♣♦✲
r❛❞❛ ❛♦s s✐st❡♠❛s ❡s❝♦❧❛r❡s ❞❛s ♥❛çõ❡s ❝♦❧♦♥✐③❛❞❛s✱ s❡ t♦r♥♦✉ ✐♥❞✐s♣❡♥sá✈❡❧ ❡♠ t♦❞♦ ♦ ♠✉♥❞♦ ❡♠ ❝♦♥s❡q✉ê♥✲ ❝✐❛ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❝✐❡♥tí✜❝♦✱ t❡❝♥♦❧ó❣✐❝♦ ❡ ❡❝♦♥ô✲ ♠✐❝♦✱ ❡ ❛✈❛❧✐❛r ❛s ❝♦♥s❡q✉ê♥❝✐❛s só❝✐♦✲❝✉❧t✉r❛✐s ❞❡ss❛ ✐♥❝♦r♣♦r❛çã♦ ✭❉✬❆▼❇❘❖❙■❖✱ ✷✵✵✵✮✳
❖ ♥♦ss♦ tr❛❜❛❧❤♦ ❜✉s❝♦✉ ❛t❡♥❞❡r ❛ t♦❞❛s ❡st❛s ✜♥❛❧✐❞❛❞❡s ❞❡s❝r✐t❛s ♣♦r ❉✬❆♠❜r♦s✐♦✱ ♥♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ♥♦s ♣r❡♦❝✉♣❛♠♦s ❝♦♠ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❧✐♠✐t❡ ❛♦ ❧♦♥❣♦ ❞❛ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳
P❡♥s❛♠♦s t❛♠❜é♠ s❡r ✐♠♣♦rt❛♥t❡ q✉❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥❤❡ç❛♠ ❛s ❞✐✜✲ ❝✉❧❞❛❞❡s ❡♥❢r❡♥t❛❞❛s ♣♦r ✈ár✐♦s ♠❛t❡♠át✐❝♦s ❛♦ ❧♦♥❣♦ ❞❛ ❤✐stór✐❛ ♣❛r❛ q✉❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ ✜❝❛ss❡ ❜❡♠ ❢✉♥❞❛♠❡♥t❛❞♦✳ ❆ ♣❛rt✐r ❞✐ss♦✱ ♦s ❡st✉❞❛♥t❡s ♣♦✲ ❞❡♠ ❡♥①❡r❣❛r ❛s s✉❛s ❞✐✜❝✉❧❞❛❞❡s ❝♦♠♦ ❛❧❣♦ ♥❛t✉r❛❧✱ q✉❡ ♣♦❞❡♠ s❡r s✉♣❡r❛❞❛s ❝♦♠ ❡s❢♦rç♦ ❡ ❞❡❞✐❝❛çã♦✳
■◆❚❘❖❉❯➬➹❖ ✶✻
❛❧❣✉♠❛s ❞❡s❝♦❜❡rt❛s ♠❛t❡♠át✐❝❛s✳ ❯♠❛ ❞❡s❝♦❜❡rt❛ ♠❛t❡♠át✐❝❛ ❡♥✈♦❧✈❡ ❝♦♥✲ tr✐❜✉✐çõ❡s ❞❡ ♦✉tr❛s ♣❡ss♦❛s ❛❧é♠ ❞❛s ❝♦♥tr✐❜✉✐çõ❡s ❞♦ ♣ró♣r✐♦ ✏❞❡s❝♦❜r✐❞♦r✑✳ ❆❧❣✉♠❛s ❞❡s❝♦❜❡rt❛s ♠❛t❡♠át✐❝❛s ♥❡♠ ❞❡✈❡r✐❛♠ s❡r ❛tr✐❜✉í❞❛s ❛ ❝❡rt♦s ♠❛✲ t❡♠át✐❝♦s✱ ♣♦✐s ❡st❛s ❥á t❡r✐❛♠ s✐❞♦ ❞❡s❝♦❜❡rt❛s ❛♥t❡r✐♦r♠❡♥t❡✳ ❊①❡♠♣❧♦ ❞✐ss♦ é ♦ Pr✐♥❝í♣✐♦ ❛tr✐❜✉í❞♦ ❛ ❈❛✈❛❧✐❡r✐✱ q✉❡ ❥á t✐♥❤❛ s❡✉s ❝♦♥❝❡✐t♦s ❝♦♥❤❡❝✐❞♦s ♣❡✲ ❧♦s ❝❤✐♥❡s❡s ♣♦r ✈♦❧t❛ ❞♦ sé❝✉❧♦ ❱■✱ ♠❡s♠♦ ♥ã♦ ✉t✐❧✐③❛♥❞♦ ❛s ✐❞❡✐❛s r❡❧❛t✐✈❛s ❛♦s ✐♥❞✐✈✐sí✈❡✐s✳ ❖✉tr♦ ❡①❡♠♣❧♦ é ♦ ❚r✐â♥❣✉❧♦ ❛tr✐❜✉í❞♦ ❛ P❛s❝❛❧✱ q✉❡ ❥á ❡r❛ ✉t✐❧✐③❛❞♦ ♣❡❧♦s ❝❤✐♥❡s❡s ❝♦♠ ♦ ♥♦♠❡ ❞❡ ❚r✐â♥❣✉❧♦ ❞❡ ❨❛♥❣❤✉✐✳ ❆❧❣✉♥s ♠❛t❡✲ ♠át✐❝♦s✱ ❝♦♠♦ ❚❛❧❡s ❞❡ ▼✐❧❡t♦✱ P✐tá❣♦r❛s ❡ ❊✉❝❧✐❞❡s✱ ♥ã♦ t❡♠ ❛ s✉❛ ❡①✐stê♥❝✐❛ r❡❛❧♠❡♥t❡ ❝♦♠♣r♦✈❛❞❛✳ ❖ q✉❡ s❡ s❛❜❡ s♦❜r❡ ❡st❡s ♠❛t❡♠át✐❝♦s ❡ s❡✉s ❢❡✐t♦s é ♦❜t✐❞♦ ❛♣❡♥❛s ❛ ♣❛rt✐r ❞❡ t❡①t♦s ❡s❝r✐t♦s sé❝✉❧♦s ❛♣ós ❛s s✉❛s s✉♣♦st❛s ❡①✐stê♥✲ ❝✐❛s✳ ◆♦ ❝❛s♦ ❞❡ P✐tá❣♦r❛s ❡ ❞❡ ❊✉❝❧✐❞❡s✱ ♦ q✉❡ ♣♦❞❡ t❡r ❛❝♦♥t❡❝✐❞♦ é ❛ ✉♥✐ã♦ ❞❡ ✉♠ ❣r✉♣♦ ❞❡ ♠❛t❡♠át✐❝♦s ✭❛ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛ ❡ ❛ ❡s❝♦❧❛ ❡✉❝❧✐❞✐❛♥❛✮ ♣❛r❛ ❝♦♠♣✐❧❛r ❛s ❞❡s❝♦❜❡rt❛s ♠❛t❡♠át✐❝❛s ❞♦ ❣r✉♣♦ ❡♠ ♥♦♠❡ ❞❡ ✉♠❛ ú♥✐❝❛ ♣❡s✲ s♦❛ ✭t❛❧✈❡③ ✐♠❛❣✐♥ár✐❛✮✳ ❆❧❣✉♥s ♠❛t❡♠át✐❝♦s ♣♦❞❡♠ t❡r s❡ ❛♣r♦✈❡✐t❛❞♦ ❞❡ s✉❛ ❝♦♥❞✐çã♦ s♦❝✐❛❧ ❡✴♦✉ st❛t✉s ♣❛r❛ s❡ ✏❛♣♦❞❡r❛r✑ ❞❡ ❞❡t❡r♠✐♥❛❞❛s ❞❡s❝♦❜❡rt❛s ♠❛t❡♠át✐❝❛s✱ ❛♥✉❧❛♥❞♦ ❞❛ ❍✐stór✐❛✱ ♦s ✈❡r❞❛❞❡✐r♦s ✏❞❡s❝♦❜r✐❞♦r❡s✑ ✭◆❖❇❘❊✱ ✷✵✵✹✮✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❉✬❆♠❜r♦s✐♦ ✭✷✵✶✷✮✱ ❛♦ ❡st✉❞❛r ❛ ❤✐stór✐❛ ❡♠ ❣❡r❛❧✱ ✏✉♠❛ ❞❛s ❣r❛♥❞❡s ❞✐✜❝✉❧❞❛❞❡s q✉❡ ❡♥❝♦♥tr❛♠♦s é ♣r♦❝✉r❛r ✈❡r ♦ ♣❛ss❛❞♦ ❝♦♠ ♦s ♦❧❤♦s ❞♦ ♣r❡s❡♥t❡ ❡ ❜✉s❝❛r ❢♦♥t❡s q✉❡ s❛t✐s❢❛ç❛♠ ♥♦ss❛s ❤❛❜✐❧✐❞❛❞❡s ❞❡ ❛♥á❧✐s❡ ❞♦ ♣r❡s❡♥t❡✑✳ ❙❡♥❞♦ ❛ss✐♠✱ ✉♠❛ ❡s❝♦❧❤❛ ❛❞❡q✉❛❞❛ ❞❛s ❢♦♥t❡s t♦r♥❛✲s❡ ❡ss❡♥❝✐❛❧ ♣❛r❛ ❡♥t❡♥❞❡r ❛ ❤✐stór✐❛✳
❊♥t❡♥❞❡♠♦s s❡r ✐♠♣♦rt❛♥t❡ ✉t✐❧✐③❛r✱ ❞✉r❛♥t❡ ♣❡sq✉✐s❛s ❡♠ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✱ ❢♦♥t❡s ♣r✐♠ár✐❛s ✭♦r✐❣✐♥❛✐s✮ ♦✉ ❝❛s♦ ♥ã♦ s❡❥❛ ♣♦ssí✈❡❧✱ ❢♦♥t❡s ♦ ♠❛✐s ♣ró①✐♠♦ ♣♦ssí✈❡❧ ❞♦s ♦r✐❣✐♥❛✐s✳ ❈♦♠ ✐ss♦✱ ❣❛r❛♥t✐♠♦s ❛♦ ♠á①✐♠♦ ❛ ✜❞❡❧✐❞❛❞❡ ❞♦s ❞❛❞♦s ♣r♦❝✉r❛❞♦s✳ ❯♠❛ ♥♦✈❛ tr❛❞✉çã♦ ♦✉ ✉♠❛ ♥♦✈❛ ✐♥t❡r♣r❡t❛çã♦ ♣♦❞❡ tr❛③❡r ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ ♦✉tr❛ ♣❡ss♦❛ q✉❡ ♣♦❞❡ ❞✐st♦r❝❡r ❞❡ ❛❧❣✉♠❛ ♠❛♥❡✐r❛ ♦s ❞❛❞♦s ♦r✐❣✐♥❛✐s✳ ▼❛s✱ ❉✬❆♠❜r♦s✐♦ ✭✷✵✶✷✮ ♥♦s ❛❧❡rt❛ q✉❡
■◆❚❘❖❉❯➬➹❖ ✶✼
❝✐❡♥tí✜❝♦s ❡ s✐st❡♠❛s ❞❡ ❡①♣❧✐❝❛çõ❡s ❞❡ ❝✉❧t✉r❛s ❝♦♠ ♣♦✉❝❛ ê♥❢❛s❡ ♥♦ r❡❣✐str♦ ❡s❝r✐t♦✳ P♦r ❡①❡♠♣❧♦✱ ❧❡♥❞❛s ❡ ❞✐t♦s ♣♦♣✉✲ ❧❛r❡s✱ ♦s ❝r♦♥✐st❛s ❡ ❝✉❧t♦s ❡ ❛ s✐♠❜♦❧♦❣✐❛✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❞❡ ♥❛t✉r❡③❛ r❡❧✐❣✐♦s❛✱ sã♦ ✐♠♣♦rt❛♥t❡s ❢♦♥t❡s ♣r✐♠ár✐❛s✳ ❊ss❡s ♠♦❞❡❧♦s ❞❡ r❡❣✐str♦ ❛♠♣❧✐❛♠ ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡ ♥♦ss♦ ❡♥t❡♥✲ ❞✐♠❡♥t♦ ❞♦ ♣❛ss❛❞♦✳ ❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✐♥str✉♠❡♥t♦s ❞❡ ❛♥á❧✐s❡ ❞❡ss❛s ❢♦♥t❡s t❡♠ s❡ ♠♦str❛❞♦ ❞❡ ❝r❡s❝❡♥t❡ ✐♠♣♦rtâ♥✲ ❝✐❛ ♥♦ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞❡ ✉♠❛ ❤✐st♦r✐♦❣r❛✜❛ ♠❛✐s ❛❞❡q✉❛❞❛ ❛♦s ♣❛ís❡s ❡ às ❝✉❧t✉r❛s ♣❡r✐❢ér✐❝❛s ✭❉✬❆▼❇❘❖❙■❖✱ ✷✵✶✷✮✳
❆♦ ❧♦♥❣♦ ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦✱ ✉t✐❧✐③❛♠♦s✱ ♥❛ ♠❛✐♦r ♣❛rt❡ ❞❛s ✈❡③❡s✱ ❢♦♥t❡s t❡r❝✐ár✐❛s ❞❡ ❇❛r♦♥ ❡ ❇♦s ✭✶✾✽✺✮ ❡ ❞❡ ❘♦q✉❡ ❡ P✐t♦♠❜❡✐r❛ ✭✷✵✶✷✮✳ ❆♦ ❝♦♠❡♥✲ t❛r♠♦s s♦❜r❡ ♦ ❧✐✈r♦ ✏❖s ❊❧❡♠❡♥t♦s✑✱ ✉t✐❧✐③❛♠♦s ❛ ❢♦♥t❡ s❡❝✉♥❞ár✐❛ ❞❡ ■r✐♥❡✉ ❇✐❝✉❞♦✱ ♣♦r s❡ tr❛t❛r ❞❡ ✉♠❛ tr❛❞✉çã♦ ❞♦ ❣r❡❣♦ ♣❛r❛ ♦ ♣♦rt✉❣✉ês✳ ◗✉❛♥❞♦ tr❛③❡♠♦s ❛❧❣✉♥s tr❛❜❛❧❤♦s ❞❡ ❆rq✉✐♠❡❞❡s✱ ✉t✐❧✐③❛♠♦s ❛ ❢♦♥t❡ s❡❝✉♥❞ár✐❛ ❞❡ ❍❡❛t❤ ✭✶✽✾✼✮ ❡ ❛ ❢♦♥t❡ t❡r❝✐ár✐❛ ❞❡ ❘♦q✉❡ ❡ P✐t♦♠❜❡✐r❛ ✭✷✵✶✷✮✱ ❞❡✈✐❞♦ à ❞✐✲ ✜❝✉❧❞❛❞❡ q✉❡ t✐✈❡♠♦s ❞❡ ❡♥❝♦♥tr❛r ♦s ♦r✐❣✐♥❛✐s ❞❡st❡ ♠❛t❡♠át✐❝♦✳ ❏á q✉❛♥❞♦ tr❛③❡♠♦s ❛❧❣✉♥s tr❛❜❛❧❤♦s ❞❡ ❈❛✈❛❧✐❡r✐✱ ✉t✐❧✐③❛♠♦s ❛s ♣r♦♣♦s✐çõ❡s ♦r✐❣✐♥❛✐s ❡✱ ♥❛s ❞❡♠♦♥str❛çõ❡s✱ ❜✉s❝❛♠♦s ❛✉①í❧✐♦ ♥❛ ❢♦♥t❡ s❡❝✉♥❞ár✐❛ ❞❡ ❙tr✉✐❦ ✭✶✾✻✾✮✳
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡♥t❡♥❞❡r ❝♦♠♦ s❡ ❞❡s❡♥✈♦❧✈❡✉ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ ❛♦ ❧♦♥❣♦ ❞❡ ♣❛rt❡ ❞❛ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✱ ❞❡s❞❡ s✉❛s ❢♦r♠❛s ♠❛✐s r✉❞✐♠❡♥t❛r❡s✱ ♣❛ss❛♥❞♦ ♣♦r ❛❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ❡ tr❛③❡♥❞♦ t❛♠❜é♠ ❛❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❛ ▼❛t❡♠át✐❝❛ ♣r♦❞✉③✐❞❛ ♥❛ ❊✉✲ r♦♣❛ ▼❡❞✐❡✈❛❧✳ ❇✉s❝❛✲s❡ t❛♠❜é♠ ❝♦♥❤❡❝❡r ❛s ❞✐✜❝✉❧❞❛❞❡s ❡♥❢r❡♥t❛❞❛s ♣❡❧♦s ♠❛t❡♠át✐❝♦s ❞❡st❡ ♣❡rí♦❞♦ ♣❛r❛ ❡♥t❡♥❞❡r ♦s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❧✐♠✐t❡✱ t❛✐s ❝♦♠♦✿ ✏✐♥✜♥✐t❛♠❡♥t❡ ♣❡q✉❡♥♦✑✱ ✏✐♥✜♥✐t❛♠❡♥t❡ ❣r❛♥❞❡✑✱ ✐♥❞✐✈✐sí✈❡✐s✱ s♦♠❛ ❞❡ s❡q✉ê♥❝✐❛s ✐♥✜♥✐t❛s✱ ❡t❝✳
❉✐✈✐❞✐♠♦s ❡st❡ tr❛❜❛❧❤♦ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✿ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✲ ✈♦s ❞❡ ❧✐♠✐t❡ ❡♠ ❝✐✈✐❧✐③❛çõ❡s ❛♥t❡s ❞♦s ❣r❡❣♦s ✭❝❛♣ít✉❧♦ ✶✮✱ ❯♠ r❡❧❛t♦ s♦❜r❡ ❛ q✉❡stã♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ♥❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ✭❝❛♣ít✉❧♦ ✷✮✱ ❆❧❣✉♥s tr❛❜❛❧❤♦s ❞❡ ❆rq✉✐♠❡❞❡s ❡ ✉♠ ♣♦✉❝♦ ❞❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ❛♣ós ❊✉❝❧✐❞❡s ✭❝❛♣ít✉❧♦ ✸✮ ❡ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s s♦❜r❡ ❧✐♠✐t❡ ♥❛ ❊✉r♦♣❛ ▼❡❞✐❡✈❛❧ ❡ ♥♦ sé❝✉❧♦ ❳❱■■ ✭❝❛♣ít✉❧♦ ✹✮✳
■◆❚❘❖❉❯➬➹❖ ✶✽
t❡♥t❛t✐✈❛s ❞❡ ❝á❧❝✉❧♦ ♣❛r❛ ♦ ✈❛❧♦r ❞❛ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦✱ r❡❧❛❝✐♦♥❛❞❛s ✐♥❞✐r❡✲ t❛♠❡♥t❡ ❝♦♠ ❛♣r♦①✐♠❛çõ❡s ♣❛r❛ ♦ ✈❛❧♦r ❞❡ π✳
❊♠ s❡❣✉✐❞❛✱ ♥♦ ❝❛♣ít✉❧♦ ✷✱ ❢❛❧❛♠♦s s♦❜r❡ ❛❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❛ ▼❛✲ t❡♠át✐❝❛ ●r❡❣❛ ♣❛r❛ ♦s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ✏❧✐♠✐t❡✑✳ ❉❡st❛❝❛♠✲s❡✿ ❛ ❊s❝♦❧❛ P✐t❛❣ór✐❝❛ ❝♦♠ ♦s ♥ú♠❡r♦s ✜❣✉r❛❞♦s❀ ❛ ✐❞❡✐❛ ❞♦s ✏✐♥❞✐✈✐sí✈❡✐s✑ ❛ss♦❝✐❛❞❛ ❝♦♠ ✉♠ ❞✐❧❡♠❛ ♣r♦♣♦st♦ ♣♦r ❉❡♠ó❝r✐t♦❀ ♦s ♣❛r❛❞♦①♦s ❞❡ ❩❡♥ã♦✱ q✉❡ ❞✐s❝✉t❡♠ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞♦ ❡s♣❛ç♦ ❡ ❞♦ t❡♠♣♦ s❡r❡♠ ❛♠❜♦s ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s ♦✉ ♥ã♦❀ ♦ ♣r♦❜❧❡♠❛ ❞♦s ✐♥❝♦♠❡♥s✉rá✈❡✐s ❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠ ♥♦✈♦ ❝♦♥❥✉♥t♦ ♥✉♠ér✐❝♦✱ ♦ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s❀ ❛❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞♦ ❧✐✈r♦ ✏❖s ❊❧❡✲ ♠❡♥t♦s✑✱ ❞❡ ❊✉❝❧✐❞❡s❀ ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s ❞❡ ❆r✐stót❡❧❡s s♦❜r❡ ♦ ✐♥✜♥✐t♦✳
Pr♦ss❡❣✉✐♠♦s✱ ♥♦ ❝❛♣ít✉❧♦ ✸✱ ❝♦♠ ♠❛✐s ❛❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞♦s ❣r❡✲ ❣♦s ♣❛r❛ ♦s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ✏❧✐♠✐t❡✑✱ ❞❡st❛❝❛♥❞♦ ❛❧❣✉♥s tr❛❜❛❧❤♦s ❞❡ ❆rq✉✐♠❡❞❡s✱ ❡♥✈♦❧✈❡♥❞♦ ❛ q✉❛❞r❛t✉r❛ ❞♦ ❝ír❝✉❧♦ ❡ ❞❛ ♣❛rá❜♦❧❛ ❡ ❛❧❣✉♠❛s q✉❡s✲ tõ❡s r❡❢❡r❡♥t❡s à ❡s♣✐r❛❧ ❞❡ ❆rq✉✐♠❡❞❡s✳ ❚r❛③❡♠♦s t❛♠❜é♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s s♦❜r❡ r❡t❛ t❛♥❣❡♥t❡ à ✉♠ ❝ír❝✉❧♦ ❡ s♦❜r❡ r❡t❛ t❛♥❣❡♥t❡ à ✉♠❛ ❝✉r✈❛ ❡♠ ❣❡r❛❧✱ s❡❣✉✐❞❛s ❞❡ ✉♠ ♣♦ssí✈❡❧ ♠ét♦❞♦ ❞❡ ❝♦♥str✉çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ à ❡s♣✐r❛❧✱ ❢❡✐t♦ ♣♦r ❆rq✉✐♠❡❞❡s✱ q✉❡ ♣♦❞❡ t❡r ✉t✐❧✐③❛❞♦ ♦ ✏♠ét♦❞♦ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦✑✳ ❋❛❧❛✲ ♠♦s ❛✐♥❞❛✱ ❜r❡✈❡♠❡♥t❡✱ s♦❜r❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞❛s ❝✉r✈❛s ❡♠ ❣❡r❛❧✱ r❡❧❛❝✐♦♥❛♥❞♦ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛ ❝♦♠ ♦ ✈❛❧♦r ❞❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡✱ q✉❡ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❛tr❛✈és ❞❛ ❞❡r✐✈❛❞❛✳
❈❆P❮❚❯▲❖ ✶✿ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ❞❡ ❧✐♠✐t❡ ❡♠ ❝✐✈✐❧✐③❛çõ❡s ❛♥t❡s ❞♦s ●r❡❣♦s ✶✾
❈❆P❮❚❯▲❖ ✶✿ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s
♣r✐♠✐t✐✈♦s ❞❡ ❧✐♠✐t❡ ❡♠
❝✐✈✐❧✐③❛çõ❡s ❛♥t❡s ❞♦s ●r❡❣♦s
P♦r ♠❛✐s ❛♥t✐❣♦s q✉❡ s❡❥❛♠ ❛❧❣✉♥s ♦❜❥❡t♦s ✭t❛♣❡t❡s✱ ✈❛s♦s✱ ❡t❝✳✮ ♣♦❞❡✲ ♠♦s ❡♥❝♦♥tr❛r ❞❡s❡♥❤❛❞♦s ♥❡❧❡s q✉❛❞r❛❞♦s✱ r❡tâ♥❣✉❧♦s✱ ♣❛r❛❧❡❧♦❣r❛♠♦s✱ tr✐â♥✲ ❣✉❧♦s✱ ❝ír❝✉❧♦s✱ ❡t❝✳ ❆❧é♠ ❞✐ss♦✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❡st❛s ❢♦r♠❛s ❣❡♦♠étr✐❝❛s ❞❡s❡♥❤❛❞❛s ❡♠ r♦❝❤❛s✳ ❖ ❝ír❝✉❧♦ t❡✈❡ t❛♠❜é♠ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡✱ ❛ ♣❛rt✐r ❞❛ ❞❡s❝♦❜❡rt❛ ❞❛ r♦❞❛✳ ❙❡❣✉♥❞♦ ❇❛r♦♥ ❡ ❇♦s ✭✶✾✽✺✱ ✉♥✐❞❛❞❡ ✶✱ ♣✳ ✶✶✮✱ ❛♣❛✲ r❡♥t❡♠❡♥t❡✱ só❧✐❞♦s tr✐❞✐♠❡♥s✐♦♥❛✐s ❝♦♠♦ ♦ ❝✉❜♦✱ t❡tr❛❡❞r♦✱ ❝✐❧✐♥❞r♦ ❡ ❛ ❡s❢❡r❛ ❡r❛♠ ❜❡♠ ❝♦♥❤❡❝✐❞♦s ❞♦s ❛♥t✐❣♦s✳
❉❡s❞❡ ♦s t❡♠♣♦s ♠❛✐s r❡♠♦t♦s✱ r❡❣r❛s ♣❛r❡❝❡♠ t❡r s✐❞♦ ✉t✐❧✐③❛❞❛s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❞❡ r❡❣✐õ❡s ❧✐♠✐t❛❞❛s ♣♦r ❧✐♥❤❛s r❡t❛s✳ ❊st❛s r❡❣r❛s ♣❛r❡❝❡♠ t❡r s✐❞♦ ❡st❛❜❡❧❡❝✐❞❛s ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ❡❧❡♠❡♥t❛r❡s✱ ✐st♦ é✱ ❝♦r✲ t❛♥❞♦ ❢♦r♠❛s ❡ r❡❛❣r✉♣❛♥❞♦ ❛s ♣❛rt❡s ♣❛r❛ ❢♦r♠❛r ✉♠❛ ✜❣✉r❛ ♠❛✐s s✐♠♣❧❡s✳ P♦r ❡①❡♠♣❧♦✱ s❡ t✐✈❡r♠♦s ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ♦✉ ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ♦✉ ✉♠ tr✐â♥❣✉❧♦ ❡s❝❛❧❡♥♦✱ ♣♦❞❡♠♦s tr❛♥s❢♦r♠á✲❧♦s✱ ✉♠ ❛ ✉♠✱ ❛tr❛✈és ❞❡ r❡❝♦rt❡s ❡ ❞❡ ✉♠ r❡❛❣r✉♣❛♠❡♥t♦ ❝♦rr❡t♦ ❞❛s ♣❛rt❡s✱ ❡♠ ✉♠ r❡tâ♥❣✉❧♦✳ ✭❇❛r♦♥ ❡ ❇♦s✱ ✶✾✽✺✱ ✉♥✐❞❛❞❡ ✶✱ ♣✳ ✶✶✮✳ ❆ ♣❛rt✐r ❞♦ r❡tâ♥❣✉❧♦ ♣♦❞❡♠♦s ❝♦♥str✉✐r ❝♦♠ ♠❛✐s ❢❛❝✐❧✐❞❛❞❡ ✉♠ q✉❛❞r❛❞♦ ❝♦♠ ❛ ♠❡s♠❛ ár❡❛ ❞❛ ✜❣✉r❛ ♦r✐❣✐♥❛❧✳ ❆ ✜❣✉r❛ ✶✳✶ ✐❧✉str❛ ❡st❛s tr❛♥s❢♦r♠❛çõ❡s✳
❈❆P❮❚❯▲❖ ✶✿ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ❞❡ ❧✐♠✐t❡ ❡♠ ❝✐✈✐❧✐③❛çõ❡s ❛♥t❡s ❞♦s ●r❡❣♦s ✷✵
❋✐❣✉r❛ ✶✳✶✿ ❚r❛♥s❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ❡❧❡♠❡♥t❛r❡s
❋♦♥t❡✿ ❆rq✉✐✈♦ ❞♦ ❛✉t♦r
❛ ár❡❛ ❞❡ t❡rr❛s ❡ r♦❝❤❛s✳ ❍á ✈ár✐❛s ❡✈✐❞ê♥❝✐❛s ❞❡ q✉❡ ♦s ❜❛❜✐❧ô♥✐♦s ✭≈ 1600
❛✳❡✳❈✳✶✮ ❞♦♠✐♥❛✈❛♠ ❛s t❡r♥❛s ❞✐t❛s s❡r❡♠ ❞❡ P✐tá❣♦r❛s ✭♥ú♠❡r♦s a✱ b ❡ ct❛✐s q✉❡ a2 +b2 = c2✮ ❡ ♣♦❞✐❛♠ ❝❛❧❝✉❧❛r r❛í③❡s q✉❛❞r❛❞❛s ❝♦♠ ❛♣r♦①✐♠❛çõ❡s ❞❡ q✉❛✐sq✉❡r ♦r❞❡♥s✳
P❛r❛ ❝❛❧❝✉❧❛r ❛s ár❡❛s ❞❡ r❡❣✐õ❡s ❧✐♠✐t❛❞❛s ♣♦r ❝✉r✈❛s✱ ❛ ❞✐✜❝✉❧❞❛❞❡ ❡r❛ ♠❛✐♦r✳ ❊♥❝♦♥tr❛r ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❢♦✐ ✉♠ ❞♦s ♣r♦❜❧❡♠❛s q✉❡ ♠❛✐s ❛tr❛✐✉ ❛ ❛t❡♥çã♦ ❞♦s ❡st✉❞✐♦s♦s ❞❡ ✈ár✐❛s é♣♦❝❛s✳ ❆❧❣✉♠❛s ❛♣r♦①✐♠❛çõ❡s ❢♦r❛♠ ❞❛❞❛s ♣❛r❛ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦✳ ❆ ♣r✐♠❡✐r❛ ❛♣r♦①✐♠❛çã♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❇❛r♦♥ ❡ ❇♦s ✭✶✾✽✺✱ ✉♥✐❞❛❞❡ ✶✱ ♣✳ ✶✷✲✶✸✮✱ ❡♥❝♦♥tr❛❞❛ ♥❛ ♠❛✐♦r✐❛ ❞❛s ❝✐✈✐❧✐③❛çõ❡s ❛♥t✐❣❛s✱ é ❞❛❞❛ ♣♦r 3r2 ✭s❡♥❞♦ r ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦✮✳ P❛r❛ ❡♥❝♦♥tr❛r ❡st❛ ❛♣r♦①✐♠❛çã♦✱ ❝♦♥s✐❞❡r❛♠♦s Q1 ♦ q✉❛❞r❛❞♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦ r ✭✜❣✉r❛ ✶✳✷✮✳ ❖ ❧❛❞♦ ✭l✮ ❞❡Q1 é ❛ ❤✐♣♦t❡♥✉s❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❝♦♠ ❝❛t❡t♦s ♠❡❞✐♥❞♦ r✳ ▲♦❣♦✱ ♣❡❧♦ ✏❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✑✱ ♦ ❧❛❞♦ ❞❡st❡ q✉❛❞r❛❞♦ ✈❛❧❡r√2✳ ❊♥tã♦✱
❛ ár❡❛ ❞❡Q1 é ✐❣✉❛❧ ❛ l2 = (r √
2)2
= 2r2✳ ❆❣♦r❛✱ ❝♦♥s✐❞❡r❛♠♦sQ2 ♦ q✉❛❞r❛❞♦ ❝✐r❝✉♥s❝r✐t♦ ❛♦ ♠❡s♠♦ ❝ír❝✉❧♦ ✭✜❣✉r❛ ✶✳✷✮✳ ❖ ❧❛❞♦ ❞❡st❡ q✉❛❞r❛❞♦ é ✐❣✉❛❧ ❛♦ ❞✐â♠❡tr♦ ❞♦ ❝ír❝✉❧♦✱ ♦✉ s❡❥❛✱ 2r✳ ▲♦❣♦✱ s✉❛ ár❡❛ ♠❡❞❡ (2r)2
= 4r2✳ P♦r ✜♠✱ ✈❛♠♦s ❝❛❧❝✉❧❛r ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ s✐♠♣❧❡s ✭M ed✮ ❡♥tr❡ ❛s ár❡❛s ❞❡ Q1 ❡ Q2✳
✶❯s❛r❡♠♦s ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦ ❡st❛ ❛❜r❡✈✐❛çã♦ q✉❡ s✐❣♥✐✜❝❛ ✏❛♥t❡s ❞❛ ❡r❛ ❈♦♠✉♠✑ ❡ é
❈❆P❮❚❯▲❖ ✶✿ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ❞❡ ❧✐♠✐t❡ ❡♠ ❝✐✈✐❧✐③❛çõ❡s ❛♥t❡s ❞♦s ●r❡❣♦s ✷✶
❆ ♠é❞✐❛ s❡rá ❞❛❞❛ ♣♦r✿
M ed= 2r
2
+ 4r2
2 =
6r2
2 = 3r
2
■st♦ ❥✉st✐✜❝❛ ❛ ♠❡❞✐❞❛ ✉t✐❧✐③❛❞❛✱ ❡♠❜♦r❛ ✉♠❛ ♠❡❧❤♦r ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ✈❛❧♦r ❞❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ♣✉❞❡ss❡ s❡r ❡♥❝♦♥tr❛❞❛✳
❋✐❣✉r❛ ✶✳✷✿ ◗✉❛❞r❛❞♦s ✐♥s❝r✐t♦s ❡ ❝✐r❝✉♥s❝r✐t♦s ❛♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ r
❋♦♥t❡✿ ❆rq✉✐✈♦ ❞♦ ❛✉t♦r
❆✐♥❞❛ s❡❣✉♥❞♦ ❇❛r♦♥ ❡ ❇♦s ✭✶✾✽✺✱ ✉♥✐❞❛❞❡ ✶✱ ♣✳ ✶✸✮✱ ♥♦ ❊❣✐t♦ ❛♥t✐❣♦✱ ❢♦✐ ✉t✐❧✐③❛❞♦ ❛❧❣♦ ✏❡q✉✐✈❛❧❡♥t❡✑ à ❢ór♠✉❧❛ ❛❜❛✐①♦ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦ ✭❆✮✿
A= 4×
8 9r
2
✭✶✳✶✮ ❯t✐❧✐③❛r❡♠♦s ✉♠❛ ♥♦t❛çã♦ ♠♦❞❡r♥❛ ♣❛r❛ t❡♥t❛r ❡①♣❧✐❝❛r ❝♦♠♦ ❡st❛ ❝✐✲ ✈✐❧✐③❛çã♦ ❝❤❡❣♦✉ à ❡st❛ ❢ór♠✉❧❛✳ ❉❡s❡♥❤❛♠♦s ✉♠ q✉❛❞r❛♥t❡ ❝✐r❝✉❧❛r ❞❡ r❛✐♦r ❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ a ✭✜❣✉r❛ ✶✳✸✮✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ár❡❛ ❞♦ q✉❛❞r❛♥t❡ ❝✐r✲ ❝✉❧❛r s❡❥❛ ✐❣✉❛❧ à ár❡❛ ❞♦ q✉❛❞r❛❞♦ ✭a2✮✳ P❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞♦ q✉❛❞r❛♥t❡ ❝✐r❝✉❧❛r✱ ❛♣❧✐❝❛♠♦s ❛ ❢ór♠✉❧❛✱ ❝♦♠♦ ❝♦♥❤❡❝❡♠♦s ❤♦❥❡ ❡♠ ❞✐❛✱ ❞❛ ár❡❛ ❞❡ ✉♠ s❡t♦r ❝✐r❝✉❧❛r ❝♦♠ â♥❣✉❧♦ ❝❡♥tr❛❧ ❞❡90◦
✿
πr2× 90
360 =
πr2
4
❈❆P❮❚❯▲❖ ✶✿ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ❞❡ ❧✐♠✐t❡ ❡♠ ❝✐✈✐❧✐③❛çõ❡s ❛♥t❡s ❞♦s ●r❡❣♦s ✷✷
❋✐❣✉r❛ ✶✳✸✿ ◗✉❛❞r❛♥t❡ ❝✐r❝✉❧❛r
❋♦♥t❡✿ ❆rq✉✐✈♦ ❞♦ ❛✉t♦r
r✐♦r♠❡♥t❡✿
πr2
4 = 3r2
4
❈♦♠♦ ❛ ár❡❛ ❞♦ q✉❛❞r❛♥t❡ ❝✐r❝✉❧❛r é ✐❣✉❛❧ à ár❡❛ ❞♦ q✉❛❞r❛❞♦ ✭a2✮✱ t❡♠♦s q✉❡
a2 = 3r
2
4 ⇒a=
r√3 2
❈♦♥s✐❞❡r❛♠♦s√3≈1,777...= 17 9 =
16
9 ✳ ▲♦❣♦✱
a = r
√
3
2 =
r× 169
2 =
16 18r=
8
9r⇒a= 8 9r
❊♥tã♦✱ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ár❡❛ ❞♦ q✉❛❞r❛♥t❡ ❝✐r❝✉❧❛r é ✐❣✉❛❧ ❛
a2 =
8 9r
2
❈♦♠♦ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ✭A✮ ❞❡ r❛✐♦ r é ✐❣✉❛❧ ❛ q✉❛tr♦ ✈❡③❡s ❛ ár❡❛ ❞♦ q✉❛❞r❛♥t❡ ❝✐r❝✉❧❛r✱ ❝❤❡❣❛♠♦s ❡♥tã♦ à ❢ór♠✉❧❛ ✶✳✶ ♣r♦❝✉r❛❞❛
A = 4×a2 ⇒A= 4×
8 9r
2
❈❆P❮❚❯▲❖ ✶✿ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ❞❡ ❧✐♠✐t❡ ❡♠ ❝✐✈✐❧✐③❛çõ❡s ❛♥t❡s ❞♦s ●r❡❣♦s ✷✸
P♦❞❡♠♦s ❞❡s❡♥✈♦❧✈❡r ❡st❛ ❢ór♠✉❧❛ ❡ ❡♥❝♦♥tr❛r ✉♠ ❜♦❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ✈❛❧♦r ❞❡π✳
A= 4×
8 9r
2
= 4× 64 81r
2
= 256 81 r
2
= 313 81r
2 ⇒
A=πr2✱ ❝♦♠ π= 313
81 ≈3,1605
❘❡ss❛❧t❛♠♦s q✉❡ ♥❛ é♣♦❝❛ ♥ã♦ ❡r❛♠ ✉t✐❧✐③❛❞❛s ❢ór♠✉❧❛s ❡ s✐♠ r❡❣r❛s ❞❡ ❝á❧❝✉❧♦ q✉❡ ❝♦♥st❛✈❛♠ ❝♦♠♦ s❡♥❞♦ ✏✉♠❛ r❡❝❡✐t❛✑ ❣❡♦♠étr✐❝❛ ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛✳ ❚❛♠❜é♠ ❞❡st❛❝❛♠♦s q✉❡ ♦ sí♠❜♦❧♦ π só ♣❛ss♦✉ ❛ s❡r ✉t✐❧✐③❛❞♦ ❛ ♣❛rt✐r ❞♦ sé❝✉❧♦ ❳❱■■■✱ ❛♣❛r❡❝❡♥❞♦ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♥♦ tr❛❜❛❧❤♦ ❞❡ ❲✐❧❧✐❛♠ ❏♦♥❡s✱ ❞❡ ✶✼✵✻✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r ♥❛ ✜❣✉r❛ ✶✳✹✳
❋✐❣✉r❛ ✶✳✹✿ ❏♦♥❡s ✭✶✼✵✻✮✱ ♠♦str❛♥❞♦ ♦ sí♠❜♦❧♦ π✳
❋♦♥t❡✿ ❤tt♣✿✴✴❛r❝❤✐✈❡✳♦r❣
❈❆P❮❚❯▲❖ ✶✿ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ❞❡ ❧✐♠✐t❡ ❡♠ ❝✐✈✐❧✐③❛çõ❡s ❛♥t❡s ❞♦s ●r❡❣♦s ✷✹
❉❡♣♦✐s✱ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦✱ ❝♦♠ π ≈ 3,1605✱ q✉❡ é ✉♠❛ ót✐♠❛
❛♣r♦①✐♠❛çã♦✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡π ≈3,14159✳
❈❆P❮❚❯▲❖ ✷✿ ❯♠ r❡❧❛t♦ s♦❜r❡ ❛ q✉❡stã♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ♥❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ✷✺
❈❆P❮❚❯▲❖ ✷✿ ❯♠ r❡❧❛t♦ s♦❜r❡ ❛
q✉❡stã♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ♥❛
▼❛t❡♠át✐❝❛ ●r❡❣❛
◆❡st❡ ❝❛♣ít✉❧♦✱ ✈❛♠♦s ❢❛❧❛r s♦❜r❡ ❛❧❣✉♠❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ♣❛r❛ ♦s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ✏❧✐♠✐t❡✑✳ ▼✉✐t❛s ❞❡st❛s ❝♦♥tr✐❜✉✐çõ❡s ❡♥✈♦❧✈❡♠ ♦ ❧✐♠✐t❡ ❞❡ s❡q✉ê♥❝✐❛s✳
✷✳✶ ❆ ❊s❝♦❧❛ P✐t❛❣ór✐❝❛ ❡ ♦s ♥ú♠❡r♦s ✜❣✉r❛❞♦s
❯♠ ♠❛t❡♠át✐❝♦ q✉❡ ❝♦st✉♠❛ s❡r ❞❡st❛❝❛❞♦ ♥❛ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛ é P✐tá❣♦r❛s ❞❡ ❙❛♠♦s ✭≈✺✼✵✲✹✾✵ ❛✳❡✳❈✳✮✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❇❛r♦♥ ❡ ❇♦s ✭✶✾✽✺✮✱ ❡❧❡ ❢✉♥❞♦✉ ✉♠❛ ❡s❝♦❧❛ ♠íst✐❝❛ ❡ ✜❧♦só✜❝❛ ❡♠ ❈r♦t♦♥❛ ✭❝♦❧ô♥✐❛ ❣r❡❣❛ ♥❛ ♣❡♥í♥s✉❧❛ ✐tá❧✐❝❛✮✿ ❛ ❡s❝♦❧❛ P✐t❛❣ór✐❝❛✳
❙❡❣✉♥❞♦ ❇❛r♦♥ ❡ ❇♦s ✭✶✾✽✺✮✱ ♦ ♥♦ss♦ ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ ❡ss❡ ♠❛t❡♠á✲ t✐❝♦ ❡ s♦❜r❡ ♦s q✉❡ ♦ s❡❣✉✐r❛♠ ❛té ♦s t❡♠♣♦s ❞❡ P❧❛tã♦ ✭≈✸✹✼ ❛✳❡✳❈✳✮ ❜❛s❡✐❛✲s❡ ❡s♣❡❝✐❛❧♠❡♥t❡ ❡♠ ✐♥❢♦r♠❛çõ❡s ❞❡ t❡r❝❡✐r♦s ✭❛❧❣✉♠❛s ✈❡③❡s ♦❜t✐❞❛s ❛♣ós ♠✉✐t♦s sé❝✉❧♦s✮✳ ❊✱ ♠❡s♠♦ ❝♦♠ ✐♥❢♦r♠❛çõ❡s ♣♦r ❡s❝r✐t♦✱ ♥✉♥❝❛ s❡ ♦❜té♠ ❛s ♣r♦✈❛s ❡ ♦s ♠ét♦❞♦s ♦r✐❣✐♥❛✐s✳
❈❆P❮❚❯▲❖ ✷✿ ❯♠ r❡❧❛t♦ s♦❜r❡ ❛ q✉❡stã♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ♥❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ✷✻
s✉♣❡r❢í❝✐❡s ❡ ❝♦♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s✉♣❡r❢í❝✐❡s✱ só❧✐❞♦s✳ ❊ ❛ss✐♠✱ s♦♠❡♥t❡ ❝♦♠ ❡ss❡s ♥ú♠❡r♦s✱ ❡r❛ ♣♦ssí✈❡❧ ❝♦♥str✉✐r ♦ ✉♥✐✈❡rs♦✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❙♣❡✉s✐♣♦ ✭sé❝✉❧♦ ■❱ ❛✳❡✳❈✳✮ ❛♣✉❞ ❇❛r♦♥ ❡ ❇♦s ✭✶✾✽✺✱ ✉♥✐❞❛❞❡ ✶✱ ♣✳ ✶✻✮✱ ♦s ♠❛✐s ✈❡❧❤♦s s❡❣✉✐❞♦r❡s ❞❡ P✐tá❣♦r❛s ❝♦♥❤❡❝✐❛♠ ♦s ♥ú♠❡r♦s tr✐❛♥❣✉❧❛r❡s✱ q✉❛❞r❛♥❣✉❧❛r❡s✱ r❡t❛♥❣✉❧❛r❡s ❡ ♦✉tr♦s ♥ú♠❡r♦s ♣♦❧✐❣♦♥❛✐s✳ ❋✐❣✉r❛ ✷✳✺✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦s ♥ú♠❡r♦s tr✐❛♥❣✉❧❛r❡s✱ q✉❛❞r❛♥❣✉❧❛r❡s✱ ❡ r❡t❛♥✲ ❣✉❧❛r❡s
❋♦♥t❡✿ ❆rq✉✐✈♦ ❞♦ ❛✉t♦r
❊♠ ♥♦t❛çã♦ ♠♦❞❡r♥❛✱ ✈❛♠♦s ♥♦s r❡❢❡r✐r à ❡ss❡s ♥ú♠❡r♦s ❝♦♠♦ T✱ Q ❡ R✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❏á ❛s r❡♣r❡s❡♥t❛çõ❡s ❞♦ s♦♠❛tór✐♦ ❞♦s ♥ú♠❡r♦sT✱Q❡R✱ ❞❡✜♥✐r❡♠♦s ❝♦♠♦ s❡♥❞♦Tn✱Qn ❡Rn✳ ▲♦❣♦✿
Tn= 1 + 2 + 3 +...+n ⇒Tn =
n(n+ 1) 2
Qn= 1 + 4 + 9 +...+n2 ⇒Qn=
n(n+ 1)(2n+ 1) 6
Rn= 2 + 6 + 12 +...+n(n+ 1)⇒Rn=
n(n+ 1)(n+ 2) 3
P❛r❛ ❝♦♥str✉çõ❡s ❡♠ três ❞✐♠❡♥sõ❡s✱ ✉t✐❧✐③❛♠♦s ♦ s♦♠❛tór✐♦ ❞❡Tn ❡Qn
♣❛r❛ ♦❜t❡r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣✐râ♠✐❞❡s tr✐❛♥❣✉❧❛r❡s ✭STn✮ ❡ ♣✐râ♠✐❞❡s q✉❛✲
❞r❛♥❣✉❧❛r❡s ✭SQn✮✳ ❖❜t❡♠♦s t❛♠❜é♠ ❝✉❜♦s ✭Cn✮✳ ❆ ✜❣✉r❛ ✷✳✻ ❛❥✉❞❛ ❛ ❡♥✲
t❡♥❞❡r ♠❡❧❤♦r ❡st❛s ❝♦♥str✉çõ❡s✳
❆♥❛❧✐s❛♥❞♦ ♦s s♦♠❛tór✐♦s✱ t❡♠♦s q✉❡✿
ST1 =T1 = 1 ST2 =T1+T2 = 1 + 3 = 4
❈❆P❮❚❯▲❖ ✷✿ ❯♠ r❡❧❛t♦ s♦❜r❡ ❛ q✉❡stã♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ♥❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ✷✼
❋✐❣✉r❛ ✷✳✻✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❛s ♣✐râ♠✐❞❡s tr✐❛♥❣✉❧❛r❡s✱ q✉❛❞r❛♥❣✉❧❛r❡s ❡ ❞♦s ❝✉❜♦s
❋♦♥t❡✿ ❇❛r♦♥ ❡ ❇♦s
SQ1 =S1 = 1 SQ2 =S1+S2 = 1 + 4 = 5
SQ3 =S1+S2+S3 = 1 + 4 + 9 = 14
C1 = 13 = 1 C2 = 23 = 8 C3 = 33 = 27
❙❡❣✉♥❞♦ ❇❛r♦♥ ❡ ❇♦s ✭✶✾✽✺✮✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦s ✜❣✉r❛❞♦s✱ r❡❧❛❝✐✲ ♦♥❛❞♦ ❛ ❝♦❧❡çõ❡s ❞❡ ✉♥✐❞❛❞❡s ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❛rr✉♠❛❞❛s ❡ ❞✐s♣♦♥í✈❡✐s ♣❛r❛ s❡ ❢♦r♠❛r ♦✉tr❛s ❝♦❧❡çõ❡s✱ ❝♦♥t✐♥❤❛ ❛ s❡♠❡♥t❡ ❞❛ ✐❞❡✐❛ ❞♦s ✏✐♥❞✐✈✐sí✈❡✐s✑ q✉❡ ✈✐r✐❛ ❛ s❡r ❡①♣❧♦r❛❞❛ ♥♦ sé❝✉❧♦ ❳❱■■✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♣♦r ❈❛✈❛❧✐❡r✐✳ ■st♦ ♦❝♦rr❡ ♣♦✐s✱ ❝♦♠ ❛ ✉♥✐ã♦ ❞❡ ✐♥✜♥✐t♦s ♣♦♥t♦s ♣♦❞❡♠♦s ❢♦r♠❛r r❡t❛s✱ ❝♦♠ ❛ ✉♥✐ã♦ ❞❡ ✐♥✜♥✐t❛s r❡t❛s ♣♦❞❡♠♦s ❢♦r♠❛r s✉♣❡r❢í❝✐❡s ❡ ❝♦♠ ❛ ✉♥✐ã♦ ❞❡ ✐♥✜♥✐t❛s s✉♣❡r❢í✲ ❝✐❡s ✭q✉❡ s❡r✐❛♠ ♦s ✏✐♥❞✐✈✐sí✈❡✐s✑✮ ♣♦❞❡♠♦s ❢♦r♠❛r só❧✐❞♦s✳ ❆tr❛✈és ❞❡st❡ t✐♣♦ ❞❡ ❝♦♥str✉çã♦✱ ♣♦❞❡♠♦s ❢♦r♠❛r q✉❛❧q✉❡r só❧✐❞♦ ❡✱ ♦ ✏❧✐♠✐t❡✑ ❞♦ s♦♠❛tór✐♦ ❞❡ t♦❞♦s ♦s só❧✐❞♦s ❡①✐st❡♥t❡s s❡rá ✐❣✉❛❧ ❛ t♦❞♦ ✉♥✐✈❡rs♦✳
✷✳✷ ❖s ✐♥❞✐✈✐sí✈❡✐s
❈❆P❮❚❯▲❖ ✷✿ ❯♠ r❡❧❛t♦ s♦❜r❡ ❛ q✉❡stã♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ♥❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ✷✽
♥❛ ❚rá❝✐❛ ✭≈ ✹✸✵ ❛✳❡✳❈✳✮✱ ✉♠❛ ❣r❛♥❞❡ ❧✐❣❛çã♦ ❝♦♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦s ❞♦s ♣✐t❛❣ór✐❝♦s✳ ▲❡✉❝✐♣♦ ❢♦✐ q✉❡♠ ❢♦r♠✉❧♦✉ ✐♥✐❝✐❛❧♠❡♥t❡ ❛ ✏t❡♦r✐❛ ❛tô♠✐❝❛✑✳
❆ ♣❛❧❛✈r❛ át♦♠♦ ✈❡♠ ❞♦ ❣r❡❣♦ ❡ s✐❣♥✐✜❝❛ ✏✐♥❞✐✈✐sí✈❡❧✑✳ ◆❛ t❡♦r✐❛ ❛tô✲ ♠✐❝❛✱ ♦ át♦♠♦ ❡r❛ ✈✐st♦ ❝♦♠♦ ❛ ♠❡♥♦r ♣❛rtí❝✉❧❛ ❞❛ ♠❛tér✐❛✱ ♥ã♦ ♣♦❞❡♥❞♦ s❡r ❞✐✈✐❞✐❞♦✳ P♦❞❡♠♦s ❡♥t❡♥❞❡r ✐st♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ❡s❝♦❧❤❡♠♦s ✉♠ ♦❜❥❡t♦ q✉❛❧q✉❡r ❡ ♦ ❞✐✈✐❞✐♠♦s ❡♠ ♣❛rt❡s ❝❛❞❛ ✈❡③ ♠❡♥♦r❡s✱ ❛té q✉❡ ❡❧❡ ♥ã♦ ♣♦ss❛ ♠❛✐s s❡r ❞✐✈✐❞✐❞♦✳ ❊st❡ s❡r✐❛ ♦ át♦♠♦✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❇❛r♦♥ ❡ ❇♦s ✭✶✾✽✺✱ ✉♥✐❞❛❞❡ ✶✱ ♣✳ ✷✵✮✱ ❉❡♠ó❝r✐t♦ ✭≈ ✹✵✵ ❛✳❡✳❈✳✮✱ ❞✐s❝í♣✉❧♦ ❞❡ ▲❡✉❝✐♣♦✱ ♥ã♦ ❛❝❡✐t❛✈❛ ❛ ✐❞❡✐❛ ❞♦s ✐♥❞✐✈✐sí✈❡✐s ❞❛ ♠❛✲ t❡♠át✐❝❛✳ ❙❡❣✉♥❞♦ ❇♦②❡r ✭✷✵✶✵✱ ♣✳ ✺✾✮✱ ♦s ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s q✉❡ ♠❛✐s ✐♥t❡r❡ss❛✈❛♠ ❛ ❉❡♠ó❝r✐t♦ ❡r❛♠ ❛q✉❡❧❡s q✉❡ ❡①✐❣✐❛♠ ❛❧❣✉♠❛ ❢♦r♠❛ ❞❡ tr❛t❛✲ ♠❡♥t♦ ✐♥✜♥✐t❡s✐♠❛❧✳ ❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛ ✭≈ ✷✽✼✲✷✶✷ ❛✳❡✳❈✳✮ ❛tr✐❜✉✐✉ à ❉❡♠ó❝r✐t♦ ♣r♦♣♦s✐çõ❡s ✐♠♣♦rt❛♥t❡s✱ t❛✐s ❝♦♠♦✿
✐✮ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ é ✐❣✉❛❧ ❛ 1
3 ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣r✐s♠❛ ❞❡ ♠❡s♠❛
❜❛s❡ ❡ ❛❧t✉r❛✳
✐✐✮ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝♦♥❡ é ✐❣✉❛❧ ❛ 1
3 ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✐❧✐♥❞r♦ ❞❡ ♠❡s♠❛
❜❛s❡ ❡ ❛❧t✉r❛✳
P❧✉t❛r❝♦ ✭≈ ✹✻✲✶✷✵✮ ❛♣✉❞ ❇❛r♦♥ ❡ ❇♦s ✭✶✾✽✺✱ ✉♥✐❞❛❞❡ ✶✱ ♣✳ ✷✵✮✱ ❤✐st♦✲ r✐❛❞♦r ❡ ❣❡ó❣r❛❢♦ ❣r❡❣♦✱ ❝✐t❛ ✉♠ ❞✐❧❡♠❛ ♣r♦♣♦st♦ ♣♦r ❉❡♠ó❝r✐t♦ ❡♠ r❡❧❛çã♦ às s❡çõ❡s ♣❛r❛❧❡❧❛s ❞❡ ✉♠ ❝♦♥❡✳
❙❡ ❝♦rt❛♠♦s ✉♠ ❝♦♥❡ ♣♦r ✉♠ ♣❧❛♥♦ ♣❛r❛❧❡❧♦ à ❜❛s❡ ✭♣❧❛♥♦ ❜❡♠ ♣ró①✐♠♦ à ❜❛s❡✮✱ ♦ q✉❡ ♣♦❞❡♠♦s ❞✐③❡r ❞❛s s✉♣❡r❢í❝✐❡s q✉❡ ❢♦r♠❛♠ ❛s s❡çõ❡s❄ ❊❧❛s sã♦ ✐❣✉❛✐s ♦✉ ❞✐❢❡r❡♥t❡s❄ ❙❡ ❡❧❛s sã♦ ❞✐❢❡r❡♥t❡s✱ ❡❧❛s t♦r♥❛rã♦ ♦ ❝♦♥❡ ✐rr❡❣✉❧❛r✱ ❝❤❡✐♦ ❞❡ ❞❡♥t❡s✱ ❝♦♠♦ ❞❡❣r❛✉s✱ ❡ ✐♠♣❛r✐❞❛❞❡s❀ ♠❛s s❡ ❡❧❛s sã♦ ✐❣✉❛✐s✱ ❛s s❡çõ❡s s❡rã♦ ✐❣✉❛✐s✱ ❡ ♣❛r❡❝❡ q✉❡ ♦ ❝♦♥❡ t❡rá ❛ ♣r♦♣r✐❡✲ ❞❛❞❡ ❞♦ ❝✐❧✐♥❞r♦ ❞❡ s❡r ❝♦♥st✐t✉í❞♦ ♣♦r ❝ír❝✉❧♦s ✐❣✉❛✐s ❡ ♥ã♦ ❞✐❢❡r❡♥t❡s✿ ♦ q✉❡ é ✉♠ ❣r❛♥❞❡ ❛❜s✉r❞♦✳
❈❆P❮❚❯▲❖ ✷✿ ❯♠ r❡❧❛t♦ s♦❜r❡ ❛ q✉❡stã♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ♥❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ✷✾
❋✐❣✉r❛ ✷✳✼✿ ❍❡❛t❤ ✭✶✾✷✶✮✱ ❝✐t❛♥❞♦ ♦ ❞✐❧❡♠❛ ❞❡ ❉❡♠ó❝r✐t♦
❋♦♥t❡✿ ❤tt♣✿✴✴❛r❝❤✐✈❡✳♦r❣
✐✮ ❙❡ ♦s ♣❧❛♥♦s sã♦ ❞✐❢❡r❡♥t❡s ❡♥tr❡ s✐✱ ❡♥tã♦ ♦ ❝♦♥❡ ♥❛ ✈❡r❞❛❞❡ é ❢♦r♠❛❞♦ ♣♦r ✐♥ú♠❡r❛s ❢❛t✐❛s ✜♥❛s ❝♦♠ ❞✐â♠❡tr♦s ❞✐❢❡r❡♥t❡s✳ ■st♦ ❞á ❛♦ ❝♦♥❡ ✉♠ ❢♦r♠❛t♦ ❝♦♠ ❞❡❣r❛✉s✱ ♥ã♦ ✉♠ ❢♦r♠❛t♦ ❧✐s♦ ❝♦♠♦ ✐♠❛❣✐♥❛♠♦s✳ P♦rt❛♥t♦✱ ♥ã♦ s❡ tr❛t❛ ❞❡ ✉♠ ❝♦♥❡✳
✐✐✮ ❙❡ ♦s ♣❧❛♥♦s sã♦ ✐❣✉❛✐s ❡♥tr❡ s✐✱ ❡♥tã♦ ♦ ❝♦♥❡✱ ♥❛ ✈❡r❞❛❞❡✱ t❡♠ ❛ ♣r♦♣r✐✲ ❡❞❛❞❡ ❞❡ ✉♠ ❝✐❧✐♥❞r♦✳
◆♦s ❞♦✐s ❝❛s♦s t❡♠♦s ✉♠ ❛❜s✉r❞♦ ❡✱ ♣♦rt❛♥t♦✱ ♦ ❝♦♥❡ ♥ã♦ ♣♦❞❡ ♥❡♠ ❡①✐st✐r✳
❆♦ ❛♥❛❧✐s❛r♠♦s ♦ ❞✐❧❡♠❛✱ ❞❡✈❡♠♦s ❞❡✐①❛r ❞❡ ❧❛❞♦ ♦ ❝♦♥❝❡✐t♦ ♠♦❞❡r♥♦ ❞❡ ❧✐♠✐t❡ ❡ ❡✈✐t❛r q✉❛❧q✉❡r ♥♦çã♦ ❞♦ ✏✐♥✜♥✐t❛♠❡♥t❡ ♣❡q✉❡♥♦✑ ❡ ❞♦ ✏✐♥✜♥✐t❛♠❡♥t❡ ❣r❛♥❞❡✑✳
❈❆P❮❚❯▲❖ ✷✿ ❯♠ r❡❧❛t♦ s♦❜r❡ ❛ q✉❡stã♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ♥❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ✸✵
✷✳✸ ❆❧❣✉♥s ♣❛r❛❞♦①♦s ❞❡ ❩❡♥ã♦
❩❡♥ã♦ ❞❡ ❊❧❡✐❛ ✭≈ ✹✻✵ ❛✳❡✳❈✳✮ ✉t✐❧✐③♦✉✲s❡ ❞♦ ❡s♣❛ç♦ ❡ ❞♦ t❡♠♣♦ ♣❛r❛ ♣r♦♣♦r s❡✉s ♣❛r❛❞♦①♦s✳ ❊♠ s❡✉ ❛r❣✉♠❡♥t♦✱ ❤❛✈❡r✐❛ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♦ ❡s♣❛ç♦ ❡ ♦ t❡♠♣♦✿
✐✮ s❡r❡♠ ❛♠❜♦s ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s❀
✐✐✮ ♥ã♦ s❡r❡♠ ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s ✭✐♥❞✐✈✐sí✈❡✐s✮✳
◆♦s ❞♦✐s ♣r✐♠❡✐r♦s ♣❛r❛❞♦①♦s ❛♣r❡s❡♥t❛❞♦s ✭❉✐❝♦t♦♠✐❛ ❡ ❆q✉✐❧❡s✮✱ s✉✲ ♣♦♠♦s ♦ ❡s♣❛ç♦ ❡ ♦ t❡♠♣♦ ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s✳ ❏á ♥♦s ❞♦✐s ú❧t✐♠♦s ♣❛r❛✲ ❞♦①♦s ❛♣r❡s❡♥t❛❞♦s ✭❋❧❡❝❤❛ ❡ ❊stá❞✐♦✮✱ s✉♣♦♠♦s q✉❡ ♦ ❡s♣❛ç♦ ❡ ♦ t❡♠♣♦ ♥ã♦ sã♦ ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s✳ ❱❡❥❛♠♦s ♦s ♣❛r❛❞♦①♦s ❛ s❡❣✉✐r✿
• ❉✐❝♦t♦♠✐❛ ✲ ❆ s✉♣♦s✐çã♦ é ❞❡ q✉❡ ♦ ❡s♣❛ç♦ ❡ ♦ t❡♠♣♦ sã♦ ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s✳ P♦rt❛♥t♦✱ ♣❛r❛ q✉❡ ♣♦ss❛♠♦s ✐r ❞❡ ✉♠ ♣♦♥t♦ ❆ ❛té ✉♠ ♣♦♥t♦ ❇ ✭s❡♥❞♦ ❆ ❡ ❇ q✉❛✐sq✉❡r✮✱ ❞❡✈❡♠♦s ♣r✐♠❡✐r♦ ❝❤❡❣❛r ❛té ❈✱ s❡♥❞♦ ❈ ❛ ♠❡t❛❞❡ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❆ ❡ ❇✳ ▼❛s ♣❛r❛ ❝❤❡❣❛r ❛té ❈✱ ❞❡✈❡♠♦s ♣r✐♠❡✐r♦ ❝❤❡❣❛r ❛té ❉✱ s❡♥❞♦ ❉ ❛ ♠❡t❛❞❡ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❆ ❡ ❈✳ ❙❡✲ ❣✉✐♥❞♦ ❡st❡ ❛r❣✉♠❡♥t♦ ✏✐♥✜♥✐t❛♠❡♥t❡✑✱ ❝❤❡❣❛♠♦s à ❝♦♥❝❧✉sã♦ ❞❡ q✉❡ ♦ ♠♦✈✐♠❡♥t♦ ❡♥tr❡ ❆ ❡ ❇ s❡r✐❛ ✐♠♣♦ssí✈❡❧✳ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❡st❡ ♣❛r❛✲ ❞♦①♦ ❝♦♠♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ t❡r♠♦s ❡♠ ✉♠❛ Pr♦❣r❡ssã♦ ●❡♦♠étr✐❝❛ ❞❡ r❛③ã♦ 1
2 ❝♦♠ ♣r✐♠❡✐r♦ t❡r♠♦ ✐❣✉❛❧ ❛d✭❞✐stâ♥❝✐❛ ❡♥tr❡ ❆ ❡ ❇✮✳ ◆❡st❡ ❝❛s♦✱
❛ s❡q✉ê♥❝✐❛ s❡r✐❛ ❛ s❡❣✉✐♥t❡✿ d,d
2,
d
4,
d
8, ...,
d
n✳ ❙❡ ❛♣❧✐❝❛r♠♦s ♦ ❝♦♥❝❡✐t♦ ♠♦❞❡r♥♦ ❞❡ ❧✐♠✐t❡ ❝♦♠ nt❡♥❞❡♥❞♦ ❛ ∞✱ ♦✉ s❡❥❛✱ lim
n→+∞ d
n✱ t❡r❡♠♦s ❝♦♠♦ r❡s✉❧t❛❞♦ ♦ ✈❛❧♦r0✱ ❝♦♥✜r♠❛♥❞♦ q✉❡ ♦ ♠♦✈✐♠❡♥t♦ ❡♥tr❡ ❆ ❡ ❇ ✏t❡♥❞❡✑ ❛
♥ã♦ ❡①✐st✐r✳ ▼❛s✱ s❛❜❡♠♦s q✉❡ ♦ ♠♦✈✐♠❡♥t♦ ❡♥tr❡ ❆ ❡ ❇ é ♣♦ssí✈❡❧✳ ❊♥✲ tã♦✱ ❛ s✉♣♦s✐çã♦ ✐♥✐❝✐❛❧ é ❢❛❧s❛ ❡ ♦ ❡s♣❛ç♦ ❡ ♦ t❡♠♣♦ ♥ã♦ sã♦ ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s✳
❈❆P❮❚❯▲❖ ✷✿ ❯♠ r❡❧❛t♦ s♦❜r❡ ❛ q✉❡stã♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ♥❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ✸✶
❋✐❣✉r❛ ✷✳✽✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ♣❛r❛❞♦①♦ ❞❛ ❉✐❝♦t♦♠✐❛
❋♦♥t❡✿ ❆rq✉✐✈♦ ❞♦ ❛✉t♦r
✉♠ ♣♦♥t♦ ❆ ❛té ✉♠ ♣♦♥t♦ ❇ ✭s❡♥❞♦ ❆ ❡ ❇ q✉❛✐sq✉❡r✮✳ ❈♦♠♦ ❆q✉✐❧❡s é✱ s✉♣♦st❛♠❡♥t❡✱ ♠❛✐s ✈❡❧♦③ q✉❡ ❛ t❛rt❛r✉❣❛✱ ❡❧❡ ❢♦✐ ❣❡♥❡r♦s♦ ❡ ❞❡✐①♦✉ ❛ t❛rt❛r✉❣❛ ❝♦♠❡ç❛r ❛ ❝♦rr✐❞❛ à s✉❛ ❢r❡♥t❡✱ ♥✉♠ ♣♦♥t♦ ❈✱ ❡♥tr❡ ❆ ❡ ❇ ✭■♥st❛♥t❡ ✵✮✳ ❈♦♠❡ç❛❞❛ ❛ ❝♦rr✐❞❛✱ q✉❛♥❞♦ ❆q✉✐❧❡s ❛❧❝❛♥ç❛r ♦ ♣♦♥t♦ ❈✱ ❛ t❛rt❛r✉❣❛ ❥á t❡rá ❝❛♠✐♥❤❛❞♦ ✉♠ ♣♦✉❝♦ ❡ ❡st❛rá ❡♠ ✉♠ ♣♦♥t♦ ❉✱ à ❢r❡♥t❡ ❞❡ ❈ ✭■♥st❛♥t❡ ✶✮✳ ◗✉❛♥❞♦ ❆q✉✐❧❡s ❛❧❝❛♥ç❛r ♦ ♣♦♥t♦ ❉✱ ❛ t❛rt❛r✉❣❛ ❥á ❡st❛rá ❡♠ ✉♠ ♣♦♥t♦ ❊✱ à ❢r❡♥t❡ ❞❡ ❉ ✭■♥st❛♥t❡ ✷✮✳ ❘❡♣❡t✐♥❞♦ ❡st❡s ❛r❣✉✲ ♠❡♥t♦s ✏✐♥✜♥✐t❛♠❡♥t❡✑✱ ❆q✉✐❧❡s ♥✉♥❝❛ ❛❧❝❛♥ç❛rá ❛ t❛rt❛r✉❣❛✱ ♣♦✐s s❡♠♣r❡ q✉❡ ❆q✉✐❧❡s ❝❤❡❣❛r ❛♦ ♣♦♥t♦ ♦♥❞❡ ❛ t❛rt❛r✉❣❛ ❡st❛✈❛ ♥♦ ✐♥st❛♥t❡ ❛♥t❡r✐♦r✱ ❛ t❛rt❛r✉❣❛ ❥á ♥ã♦ ❡st❛rá ♠❛✐s ❧á ❡ s✐♠ ❡♠ ✉♠ ♣♦♥t♦ à ❢r❡♥t❡ ❞♦ ♣♦♥t♦ ❛♥t❡r✐♦r✳ ■st♦ é ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s s❛❜❡♠♦s q✉❡ s❡ ❆q✉✐❧❡s é ♠❛✐s rá♣✐❞♦ ❞♦ q✉❡ ❛ t❛rt❛r✉❣❛✱ ❡♠ ❛❧❣✉♠ ✐♥st❛♥t❡ ❡❧❡ ✐rá ❛❧❝❛♥ç❛r ❛ t❛rt❛r✉❣❛ ❡✱ ❡♠ ♦✉tr♦ ✐♥st❛♥t❡✱ ✐rá ✉❧tr❛♣❛ssá✲❧❛✳ ❖ ❛❜s✉r❞♦ ✈❡♠ ❞❡ s✉♣♦r♠♦s q✉❡ ♦ ❡s♣❛ç♦ ❡ ♦ t❡♠♣♦ sã♦ ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s✳ ❊♥tã♦✱ ❛ s✉♣♦s✐çã♦ ✐♥✐❝✐❛❧ é ❢❛❧s❛ ❡ ♦ ❡s♣❛ç♦ ❡ ♦ t❡♠♣♦ ♥ã♦ sã♦ ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s✳
❆❣♦r❛✱ ❛ s✉♣♦s✐çã♦ é q✉❡ ♦ ❡s♣❛ç♦ ❡ ♦ t❡♠♣♦ ♥ã♦ sã♦ ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s✳ ❙❡♥❞♦ ❛ss✐♠✱ ❡①✐st❡ ✉♠❛ ♠❡♥♦r ✉♥✐❞❛❞❡ ❞❡ ❡s♣❛ç♦ ✭✉♠ ♣♦♥t♦✮ ❡ ✉♠❛ ♠❡♥♦r ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦ ✭✉♠ ✐♥st❛♥t❡✮✳
❈❆P❮❚❯▲❖ ✷✿ ❯♠ r❡❧❛t♦ s♦❜r❡ ❛ q✉❡stã♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ♥❛ ▼❛t❡♠át✐❝❛ ●r❡❣❛ ✸✷
❋✐❣✉r❛ ✷✳✾✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ♣❛r❛❞♦①♦ ❞❡ ❆q✉✐❧❡s
❋♦♥t❡✿ ❆rq✉✐✈♦ ❞♦ ❛✉t♦r
• ❊stá❞✐♦ ✲ ■♠❛❣✐♥❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❜❧♦❝♦s A✱ B ❡ C✱ t♦❞♦s ❞♦ ♠❡s♠♦ t❛♠❛♥❤♦✳ ❖s ❜❧♦❝♦s ❆ ♠♦✈❡♠✲s❡ ✉♠ ❡s♣❛ç♦ ♣❛r❛ ❛ ❞✐r❡✐t❛ ❡♠ ❝❛❞❛ ✐♥st❛♥t❡✱ ♦s ❜❧♦❝♦s B ♣❡r♠❛♥❡❝❡♠ ✐♠ó✈❡✐s ❡ ♦s ❜❧♦❝♦s C ♠♦✈❡♠✲s❡ ✉♠ ❡s♣❛ç♦ ♣❛r❛ ❛ ❡sq✉❡r❞❛ ❡♠ ❝❛❞❛ ✐♥st❛♥t❡✳ ◆♦ ✐♥st❛♥t❡ 0✱ A4 ❡stá ♣♦s✐✲
❝✐♦♥❛❞♦ ❧♦❣♦ ❛❝✐♠❛ ❞❡ C2✳ ❯♠ ✐♥st❛♥t❡ ❛♣ós ✭■♥st❛♥t❡ 1✮✱ A4 ♣❛ss❛ ❛
❡st❛r ♣♦s✐❝✐♦♥❛❞♦ ❛❝✐♠❛ ❞❡ C4✳ ▲♦❣♦✱ ❡♠ r❡❧❛çã♦ ❛♦ ❜❧♦❝♦ C✱ ♦ ❜❧♦❝♦ ❆ ♠♦✈❡✉ ❞♦✐s ❡s♣❛ç♦s ❡♠ ✉♠ ú♥✐❝♦ ✐♥st❛♥t❡✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳
❋✐❣✉r❛ ✷✳✶✵✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ♣❛r❛❞♦①♦ ❞♦ ❊stá❞✐♦ ❞❡ ❩❡♥ã♦
❋♦♥t❡✿ ❆rq✉✐✈♦ ❞♦ ❛✉t♦r
