▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛✿
❯♠❛ ❛♣❧✐❝❛çã♦ ❞✐r❡t❛ ♥♦ ❝♦t✐❞✐❛♥♦
♣♦r
❍❡r❜❡rt ❏♦sé ❈❛✈❛❧❝❛♥t✐ ❞❡ ❙♦✉③❛
▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛✿
❯♠❛ ❛♣❧✐❝❛çã♦ ❞✐r❡t❛ ♥♦ ❝♦t✐❞✐❛♥♦
†♣♦r
❍❡r❜❡rt ❏♦sé ❈❛✈❛❧❝❛♥t✐ ❞❡ ❙♦✉③❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞❛
Pr♦❢
❛✳ ❊❧✐s❛♥❞r❛ ❞❡ ❋át✐♠❛ ●❧♦ss ❞❡ ▼♦r❛❡s
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ r❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚ ✲ ❉▼ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
▼❛rç♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇
† ❊st❡ tr❛❜❛❧❤♦ ❝♦♥t♦✉ ❝♦♠ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❞❛ ❈❛♣❡s✳
❆❣r❛❞❡❝✐♠❡♥t♦s
◆ã♦ ♣♦✉❝❛s ♣❡ss♦❛s ❢♦r❛♠ ✐♠♣♦rt❛♥t❡s ♣❛r❛ q✉❡ ♠❛✐s ❡ss❛ ❡t❛♣❛ s❡ r❡❛❧✐③❛ss❡✳ ❙❡♠ ♦ ❛✉①í❧✐♦ ❞❡❧❡s✱ ♥❛❞❛ t❡r✐❛ s✐❞♦ ♣♦ssí✈❡❧✳ ❈♦♠♦ ❥á ❞✐ss❡ ♦ P♦❡t❛ ▼❛♥♦❡❧ ❞❡ ❇❛rr♦s✱ ✏❖s ❖✉tr♦s✿ ♦ ♠❡❧❤♦r ❞❡ ♠✐♠ s♦✉ ❊❧❡s✑✳ ❆ss✐♠ ✜❝❛ ♠❡✉ ❛❣r❛❞❡❝✐♠❡♥t♦ à ❈❆P❊❙✱ ❛♦s ♣r♦❢❡ss♦r❡s✱ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❡ ❛♦s ❝♦♦r❞❡♥❛❞♦r❡s✳
❉❡❞✐❝❛tór✐❛
❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛ ❉❡✉s✱ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛ ❡ ❛ t♦❞❛s ❛s ♣❡ss♦❛s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♠❡✉ s✉❝❡ss♦ ❡ ❝r❡s❝✐♠❡♥t♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s tó♣✐❝♦s ❞❛ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛✱ ❜✉s❝❛♥❞♦ s❡♠♣r❡ ❢❛③❡r ❧✐❣❛çã♦ ✐♠❡❞✐❛t❛ ❝♦♠ ❡✈❡♥t♦s ❞❡ ♥♦ss❛ r❡❛❧✐❞❛❞❡✳ P❛ss❛♠♦s ♣♦r ❛❧❣✉♥s ❛ss✉♥t♦s ♥ã♦ ❛❜♦r❞❛❞♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❝♦♠ ✐♥t✉✐t♦ ❞❡ ❢♦r♥❡❝❡r ❢❡rr❛♠❡♥t❛s ❜ás✐❝❛s ♣❛r❛ ❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦ ❡♠ ♥♦ss♦ ❝♦t✐❞✐❛♥♦✳ ❊st✉❞❛♠♦s t❛♠❜é♠ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❡❧❡trô♥✐❝❛ q✉❡ ♥♦s ❛✉①✐❧✐❛ ❛ r❡s♦❧✈❡r ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s q✉❡ ♣♦ss✉❡♠ ❡①t❡♥s♦s ❝á❧❝✉❧♦s✳
P❛❧✈r❛s✲❝❤❛✈❡✿ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛✳ ❏✉r♦s✳ ❉❡s❝♦♥t♦s✳ ❙✐st❡♠❛s ❞❡ ❆♠♦rt✐③❛çã♦✳ P❧❛♥✐❧❤❛s ❊❧❡trô♥✐❝❛s✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② t❤❡ ♠❛✐♥ t♦♣✐❝s ❛❜♦✉t ▼❛t❤❡♠❛t✐❝❛❧ ❋✐♥❛♥❝❡✱ s❡❡❦✐♥❣ ❢♦r ✐♠♠❡❞✐❛t❡ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ ♦✉r r❡❛❧✐t②✳ ❲❡ st✉❞② s♦♠❡ s✉❜❥❡❝ts ❞✐s❝✉ss❡❞ ✐♥ ❍✐❣❤ ❙❝❤♦♦❧ ✐♥ ♦r❞❡r t♦ ♣r♦✈✐❞❡ ❜❛s✐❝ t♦♦❧s ❢♦r ❞❡❝✐s✐♦♥ ♠❛❦✐♥❣ ✐♥ ♦✉r ❞❛✐❧② ❧✐✈❡s✳ ❲❡ ❛❧s♦ st✉❞✐❡❞ ❛♥ ❡❧❡❝tr♦♥✐❝ t♦♦❧ t❤❛t ❤❡❧♣s ✉s t♦ s♦❧✈❡ s❡✈❡r❛❧ ♣r♦❜❧❡♠s t❤❛t ❤❛✈❡ ❡①t❡♥s✐✈❡ ❝❛❧❝✉❧❛t✐♦♥s✳
❑❡②✇♦r❞s✿ ▼❛t❤❡♠❛t✐❝❛❧ ❋✐♥❛♥❝❡✳ ■♥t❡r❡st✳ ❉✐s❝♦✉♥t✐♥❣✳ ❆♠♦rt✐③❛t✐♦♥✳ ❙♣r❡❛❞s❤❡❡t✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ Pr❡❧✐♠✐♥❛r❡s ✹
✶✳✶ Pr♦❣r❡ssõ❡s ❆r✐t♠ét✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ Pr♦❣r❡ssõ❡s ●❡♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❘❛③ã♦ ❡ Pr♦♣♦rçã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸✳✶ ❘❛③ã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸✳✷ Pr♦♣♦rçã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸✳✸ ●r❛♥❞❡③❛s ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸✳✹ ●r❛♥❞❡③❛s ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✹ P♦r❝❡♥t❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ✷✷
✷✳✶ ❖♣❡r❛çõ❡s ❈♦♠❡r❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷ ❏✉r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷✳✶ ❚❛①❛s ❞❡ ❏✉r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷✳✷ ❘❡❣✐♠❡s ❞❡ ❈❛♣✐t❛❧✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸ ❈❛♣✐t❛❧✐③❛çã♦ ❉✐s❝r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸✳✶ ❏✉r♦s ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✸✳✷ ❏✉r♦s ❈♦♠♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✹ P❡rí♦❞♦ ❋r❛❝✐♦♥ár✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✷✳✺ ❏✉r♦s ❙✐♠♣❧❡s ×❏✉r♦s ❈♦♠♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✷✳✻ ❚✐♣♦s ❞❡ ❚❛①❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✻✳✶ ❚❛①❛s Pr♦♣♦r❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✻✳✷ ❚❛①❛s ❊q✉✐✈❛❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✻✳✸ ❚❛①❛ ❊❢❡t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✻✳✹ ❚❛①❛ ◆♦♠✐♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✻✳✺ ❖ ❡❢❡✐t♦ ❞❛ ✐♥✢❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✼ ❖♣❡r❛çõ❡s ❞❡ ❉❡s❝♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✼✳✶ ❉❡s❝♦♥t♦s ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✼✳✷ ❉❡s❝♦♥t♦s ❈♦♠♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✽ ❊q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❈❛♣✐t❛✐s ❛ ❏✉r♦s ❈♦♠♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✽✳✶ ❊q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❞♦✐s ❝❛♣✐t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✷✳✽✳✷ ❱❛❧♦r ❆t✉❛❧ ❞❡ ✉♠ ❈♦♥❥✉♥t♦ ❞❡ ❈❛♣✐t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✾ ❙✐st❡♠❛ ❞❡ ❆♠♦rt✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✷✳✾✳✶ ❖ ❙✐st❡♠❛ ❞❡ Pr❡st❛çõ❡s ❈♦♥st❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✷✳✾✳✷ ❖ ❙✐st❡♠❛ ❞❡ ❆♠♦rt✐③❛çã♦ ❈♦♥st❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✷✳✾✳✸ ❖ ❙✐st❡♠❛ ❞❡ ❆♠♦rt✐③❛çã♦ ❈r❡s❝❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✷✳✾✳✹ ❈♦♠♣❛r❛♥❞♦ ♦s s✐st❡♠❛s ❞❡ ❛♠♦rt✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✸ P❧❛♥✐❧❤❛s ❊❧❡trô♥✐❝❛s ✼✵
❆♣ê♥❞✐❝❡ ✼✾
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✽✺
■♥tr♦❞✉çã♦
❆ s♦❝✐❡❞❛❞❡ ♣ré✲❤✐stór✐❝❛ ❡r❛ ❝♦♠♣♦st❛ ♣♦r ❤♦♠❡♥s ❜r✉t♦s✱ q✉❡ ✈✐✈✐❛♠ ❡♠ ♣❡q✉❡♥♦s ❣r✉♣♦s ♥ô♠❛❞❡s✳ ❋r❡♥t❡ ❛ ❡ss❡ ❝❡♥ár✐♦✱ t♦r♥♦✉✲s❡ ❞✐❢í❝✐❧ ♦ ♣r♦❣r❡ss♦ ❞♦ ❝♦♠ér❝✐♦✳ ❙ó ❝♦♠ ♦ ♣❛ss❛r ❞♦ t❡♠♣♦✱ ❥✉♥t♦ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ té❝♥✐❝❛s ♣❛r❛ ♦ ♣❧❛♥t✐♦ ❡♠ ❣r❛♥❞❡ ❡s❝❛❧❛✱ é q✉❡ ❡❧❡s ❝♦♠❡ç❛r❛♠ ❛ ✜①❛r r❛í③❡s✱ ❞❛♥❞♦ ✐♥í❝✐♦ ❛ ♣❡q✉❡♥♦s ✈✐❧❛r❡❥♦s q✉❡ ♣r♦♣♦r❝✐♦♥❛r❛♠ ❛ ❡✈♦❧✉çã♦ ❞❛s r❡❧❛çõ❡s s♦❝✐♦❡❝♦♥ô♠✐❝❛s✳
❈♦♠ ❛ ♣r♦❞✉çã♦ ❞❡ ❛❧✐♠❡♥t♦s ❡ ♦✉tr♦s ✉t❡♥sí❧✐♦s ✐♥✐❝✐❛✲s❡ ♦ ❡s❝❛♠❜♦✱ ❝♦♥❤❡❝✐❞♦ t❛♠❜é♠ ❝♦♠♦ tr♦❝❛s ❞✐r❡t❛s✱ ♦✉ s❡❥❛✱ ✉♠ ♣r♦❞✉t♦ ♣❡❧♦ ♦✉tr♦✳
❯♠❛ ❞❛s ❝✐✈✐❧✐③❛çõ❡s ❛♥t✐❣❛s q✉❡ ♣r❛t✐❝❛✈❛♠ ❡ss❡ s✐st❡♠❛ ❡❝♦♥ô♠✐❝♦ ❡r❛ ♦s s✉♠ér✐♦s✳ ❘❡❣✐str♦s s♦❜r❡ ❡ss❡ ♣♦✈♦ ♠♦str❛♠ ♦ ❛♣♦♥t❛♠❡♥t♦ ❡♠ tá❜✉❛s ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛s q✉❡ ♣❡r♠❡✐❛♠ ❛ ♠❛t❡♠át✐❝❛ ✜♥❛♥❝❡✐r❛✱ t❛✐s ❝♦♠♦ ❥✉r♦s✱ ❡①♣♦♥❡♥❝✐❛✐s✱ s✐st❡♠❛s ❞❡ ♣❡s♦s ❡ ♠❡❞✐❞❛s✱ ❤✐♣♦t❡❝❛s✱ ❡t❝✳✱ ❛❧é♠ ❞❛s q✉❡ ❝♦♥st❛✈❛♠ r❡❧❛t♦s ❞❡ ❡♠♣r❡s❛s ✈♦❧t❛❞❛s ❛♦ ♠❡r❝❛❞♦✳
❯♠ ♣r♦❜❧❡♠❛ q✉❡ s✉r❣✐✉ ❢♦✐ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛✈❛❧✐❛r ❡ ❝❛❧❝✉❧❛r ♦s ❜❡♥s ❛ s❡r❡♠ tr♦❝❛❞♦s✱ ❢♦✐ ✐ss♦ q✉❡ ❢❡③ ❝♦♠ q✉❡ s✉r❣✐ss❡ ❛ ♠♦❡❞❛✳ ❍✐st♦r✐❝❛♠❡♥t❡✱ ❛❧❣✉♠❛s ♠❡r❝❛❞♦r✐❛s✱ ♣♦r s❡r❡♠ ♠❛✐s ❞❡♠❛♥❞❛❞❛s q✉❡ ♦✉tr❛s✱ ❡r❛♠ r❡❢❡rê♥❝✐❛s ❞❡ ✈❛❧♦r ❡✱ ♣♦r ❝♦♥s❡q✉ê♥❝✐❛✱ ❡①❡r❝✐❛♠ ❛ ❢✉♥çã♦ ❞❡ tr♦❝❛ ❞❛ ♠♦❡❞❛✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ s❛❧ ❡ ♦ ❣❛❞♦✳ ❈♦♠ ♦ ♣❛ss❛r ❞♦ t❡♠♣♦ t♦r♥❛r❛♠✲s❡ ❞✐❢í❝❡✐s ❛s tr❛♥s❛çõ❡s ❝♦♠ ❡ss❡s ♠❛t❡r✐❛✐s✱ s✉r❣✐♥❞♦ ♥♦ sé❝✉❧♦ ❱■■ ❛✳❈✳ ❛s ♣r✐♠❡✐r❛s ♠♦❡❞❛s ♠❡tá❧✐❝❛s✿ ♣❡ç❛s ❝♦♠ ♣❡s♦ ❡ ✈❛❧♦r ❞❡t❡r♠✐♥❛❞♦s✱ ❝♦♥t❡♥❞♦ ♦ ❝✉♥❤♦ ♦✜❝✐❛❧ ✐♠♣r❡ss♦ q✉❡ ❧❤❡s ❣❛r❛♥t❡♠ ♦ s❡✉ ✈❛❧♦r✳
❉❡t❡♥t♦r❡s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ ❧✉❝r♦✱ ❛s♣❡❝t♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ♠❛t❡♠át✐❝❛
✜♥❛♥❝❡✐r❛✱ ❡ ❝♦♠ ♦ ❛✉♠❡♥t♦ ❞❛ ♣r♦❝✉r❛ ♣❡❧♦s s❡r✈✐ç♦s ♦❢❡rt❛❞♦s✱ ♦s ❝♦♠❡r❝✐❛♥t❡s ❞❡ ❞✐♥❤❡✐r♦ ❝♦♠❡ç❛r❛♠ ❛ ❝♦❜r❛r ✉♠❛ ❝❡rt❛ q✉❛♥t✐❛ s♦❜r❡ ♦ ✈❛❧♦r ♥❡❣♦❝✐❛❞♦✳ ❆♦ ❡①❡r❝❡r s✉❛s ❛t✐✈✐❞❛❞❡s ❡♠ ❜❛♥❝♦s ❞❡ ♣r❛ç❛s✱ ❡ss❡s ♣r♦✜ss✐♦♥❛✐s ❡r❛♠ ✐♥t✐t✉❧❛❞♦s ❞❡ ❜❛♥q✉❡✐r♦s✱ ❡ ❝♦♠ ❛ ❡✈♦❧✉çã♦ ❞❛ ❡❝♦♥♦♠✐❛ ❡ss❛ ♣rát✐❝❛ ❢♦✐ s❡ ❛♣r✐♠♦r❛♥❞♦ ❡ s❡ ✜r♠❛♥❞♦ ❡♠ ✐♥st✐t✉✐çõ❡s ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❜❛♥❝♦s✱ q✉❡ ❞✐♥❛♠✐③❛r❛♠ ❛ ❡❝♦♥♦♠✐❛✱ ❝♦♥tr✐❜✉✐♥❞♦ ❝♦♠ ♦ s✉r❣✐♠❡♥t♦ ❞♦s ❥✉r♦s ❝♦♠♣♦st♦s ❡ ❝♦♠ ♦ ❛♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❛s té❝♥✐❝❛s ✜♥❛♥❝❡✐r❛s✳
❆ tr❛♥s❛çã♦ ❞♦ ❞✐♥❤❡✐r♦ ❡stá ✐♥tr✐♥s✐❝❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛ ❛♦ ❝ré❞✐t♦ ❡ ❛♦ t❡♠♣♦✱ ♣♦✐s é ❛tr❛✈és ❞❡❧❡s q✉❡ s❡ ❣❡r❛♠ ♦s ❥✉r♦s✳ ❆♥t❡s ❞❛ ❡①♣❛♥sã♦ ❝♦♠❡r❝✐❛❧ ❡ ❞♦ s✉r❣✐♠❡♥t♦ ❞♦ s✐st❡♠❛ ❝❛♣✐t❛❧✐st❛✱ ♦s ❥✉r♦s ❡r❛♠ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❝✉♥❤♦ ét✐❝♦✳ ❖ ❛❝ú♠✉❧♦ ❞❡ ❝❛♣✐t❛❧ ♣♦r ❛❧❣✉♥s ✐♥❞✐✈í❞✉♦s ❡ ♦ ❞é✜❝✐t ❞❡ ♦✉tr♦s ✜③❡r❛♠ s✉r❣✐r ♦ s✐st❡♠❛ ✜♥❛♥❝❡✐r♦✱ q✉❡ ❡①❡r❝❡ ❛ ❢✉♥çã♦ ❞❡ ✐♥t❡r♠❡❞✐❛❞♦r ❡♥tr❡ ♦s r❡❝✉rs♦s ❞♦s ♣♦✉♣❛❞♦r❡s ❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝❛♣t❛çã♦ ❞♦s ❞❡✜❝✐tár✐♦s✳ ❊ss❛ ♣❡r♠✉t❛ ❞❡ r❡❝✉rs♦s ❣❡r❛ ❧✉❝r♦ ♣♦r ♠❡✐♦ ❞❡ ❥✉r♦s q✉❡ ♣❡r♠❡✐❛♠ ❛s ✐♥t❡r♠❡❞✐❛çõ❡s ✜♥❛♥❝❡✐r❛s✳
❚♦♠❛♥❞♦ ♦ tr✐♣é ✲ t❡♠♣♦✱ ❝ré❞✐t♦ ❡ ❥✉r♦s ✲ ❝♦♠♦ ❢❛t♦r❡s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❛ ❡❝♦♥♦♠✐❛ ❡ ❞❛ s♦❝✐❡❞❛❞❡✱ ♣❡r❝❡❜❡✲s❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ tr❛③❡r à ❧✉③ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❝❛♣❛③ ❞❡ ❡st❛❜❡❧❡❝❡r ❝♦♥❝❡✐t♦s ❡ r❡❣r❛s q✉❡ ♣♦ss✐❜✐❧✐t❡♠ ♦ ❡st✉❞♦ ❞❛ ✈❛r✐❛çã♦ ❞♦ ❞✐♥❤❡✐r♦ ❛tr❛✈és ❞♦s t❡♠♣♦s✿ ❛ ♠❛t❡♠át✐❝❛ ✜♥❛♥❝❡✐r❛✳ ❊ss❛ ✈❡rt❡♥t❡ ❞❛ ♠❛t❡♠át✐❝❛ ❞❡✈❡ s❡r ❝♦♥s✐❞❡r❛❞❛✱ t❡♥❞♦ ❡♠ ✈✐st❛ ❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ✐♥❝r❡♠❡♥t♦ ❞♦ s❡♥s♦ ❝rít✐❝♦ ❞♦s ❛❧✉♥♦s ❢r❡♥t❡ ❛♦ s✐st❡♠❛ ❡❝♦♥ô♠✐❝♦ ❡♠ q✉❡ ❡stã♦ ✐♥s❡r✐❞♦s✱ r❡s✉❧t❛♥❞♦ ♥❛ ❢♦r♠❛çã♦ ❞❡ ❝✐❞❛❞ã♦s ❝♦♥s❝✐❡♥t❡s ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ às ✜♥❛♥ç❛s ♣❡ss♦❛✐s✳
◆♦ ♠✉♥❞♦ ❝♦♥t❡♠♣♦râ♥❡♦ ❛s r❡❧❛çõ❡s ✜♥❛♥❝❡✐r❛s ❢❛③❡♠ ♣❛rt❡ ❞❛s ❛t✐✈✐❞❛❞❡s ❝♦t✐❞✐❛♥❛s✱ ❡ ♣♦r ❡ss❡ ♠♦t✐✈♦ é ✐♠♣♦rt❛♥t❡ ❛❞q✉✐r✐r ♥♦çõ❡s ❞❡ ❝♦♠♦ ❛❞♠✐♥✐str❛r ♦ ❞✐♥❤❡✐r♦✳
❆ ✜♠ ❞❡ ♣❧❛♥❡❥❛r ❣❛st♦s ❡ ❝r✐❛r r❡❧❛çõ❡s ❞❡ ❝♦♥s✉♠♦ ❝♦♠ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡✱ é ❢✉♥❞❛♠❡♥t❛❧ q✉❡ ♦ ✐♥❞✐✈í❞✉♦ ❡♥t❡♥❞❛ ♦s ❝♦♥❝❡✐t♦s ✜♥❛♥❝❡✐r♦s ❞❡ t❛❧ ♠❛♥❡✐r❛
q✉❡ ♣♦ss❛ ❞❡s❡♥✈♦❧✈❡r ✈❛❧♦r❡s ❡ ❝♦♠♣❡tê♥❝✐❛s ♥❡❝❡ssár✐❛s ♣❛r❛ q✉❡ s❡ t♦r♥❡ ♠❛✐s ❝♦♥s❝✐❡♥t❡ ❞❛s ♦♣♦rt✉♥✐❞❛❞❡s ❡ r✐s❝♦s ♥❡❧❡s ❡♥✈♦❧✈✐❞♦s✳
❆ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ s❡ ❢❛③ ♠✉✐t♦ ♣r❡s❡♥t❡✱ ♣♦st♦ q✉❡ ❛ ♠❡s♠❛ t❡♠ ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s✱ ♥ã♦ só ♥♦ ❞✐❛ ❛ ❞✐❛ ❞❛ ♣♦♣✉❧❛çã♦ ❡♠ ❣❡r❛❧✱ ♠❛s ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ ❝♦t✐❞✐❛♥♦ ❞♦s ❣❡st♦r❡s ❡ ♣r♦✜ss✐♦♥❛✐s q✉❡ ♥❡❝❡ss✐t❛♠ ❞❛ ♠❡s♠❛ ♣❛r❛ ✜♥s ❞❡ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦✳ ➱ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❡ss❡♥❝✐❛❧ ♣❛r❛ ♦ ❣❡r❡♥❝✐❛♠❡♥t♦ ❡✜❝✐❡♥t❡ ❡ s✉❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ é ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛✱ ♣♦✐s tr❛③ ♠❛✐♦r r❡♥t❛❜✐❧✐❞❛❞❡ ❡ ♠❛①✐♠✐③❛çã♦ ❞♦s ❧✉❝r♦s✳
❊st❡ tr❛❜❛❧❤♦ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ♠♦str❛r ❛♦s ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ❡st✉❞♦ ❡ ❝♦♠♣r❡❡♥sã♦ ❞❛ ♠❛t❡♠át✐❝❛ ✜♥❛♥❝❡✐r❛ ♣❛r❛ ❛s t♦♠❛❞❛s ❞❡ ❞❡❝✐sõ❡s ❞❡ ❢♦r♠❛ r❛❝✐♦♥❛❧✳ ❊♠ ✈✐st❛ ❞✐st♦✱ ♦ tr❛❜❛❧❤♦ é ❝♦♠♣♦st♦ ♣♦r três ❝❛♣ít✉❧♦s✳ ◆♦ ❈❛♣ít✉❧♦ 1 ❞❛♠♦s ❛s ❢❡rr❛♠❡♥t❛s ♥❡❝❡ssár✐❛s ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❡
♠❛t❡♠át✐❝❛ ✜♥❛♥❝❡✐r❛✱ ❝♦♠♦ s❡q✉ê♥❝✐❛s✱ ♣r♦♣♦rçã♦ ❡ ♣♦r❝❡♥t❛❣❡♠✳ ◆♦ ❈❛♣ít✉❧♦ 2
sã♦ ✐♥tr♦❞✉③✐❞♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ❥✉r♦s s✐♠♣❧❡s ❡ ❝♦♠♣♦st♦s✱ ❛❧é♠ ❞❡ ❝♦♥t❡ú❞♦s ♠❛✐s s♦✜st✐❝❛❞♦s ❝♦♠♦ é ♦ ❝❛s♦ ❞❡ ❛♠♦rt✐③❛çã♦ ❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❝❛♣✐t❛✐s✳ ❖ ❈❛♣ít✉❧♦3é
❝♦♠♣♦st♦ ♣♦r r❡s♦❧✉çõ❡s ❞❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ♣❧❛♥✐❧❤❛s ❡❧❡trô♥✐❝❛s ❝♦♠ t♦❞♦s ♦s ❝♦♥t❡ú❞♦s ♠❡♥❝✐♦♥❛❞♦s ♥♦ tr❛❜❛❧❤♦✳ ❋✐♥❛❧✐③❛♠♦s ❝♦♠ ✉♠ ❛♣ê♥❞✐❝❡ tr❛③❡♥❞♦ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♠❛✐s ❣❡r❛✐s s♦❜r❡ s❡q✉ê♥❝✐❛s ❞❡ ♥ú♠❡r♦s r❡❛✐s✳
❈❛♣ít✉❧♦ ✶
Pr❡❧✐♠✐♥❛r❡s
◆❡st❡ ❈❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ♥❡❝❡ssár✐♦s à ❝♦♠♣r❡❡♥sã♦ ❞❡ ♥♦ss❛ ❞✐ss❡rt❛çã♦✳ ❊♥✉♥❝✐❛r❡♠♦s ❡ ♣r♦✈❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ❡ ❣❡♦♠étr✐❝❛s ❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❛ ▼❛t❡♠át✐❝❛ ❇ás✐❝❛✱ t❛✐s ❝♦♠♦ r❛③ã♦✱ ♣r♦♣♦rçã♦ ❡ ♣♦r❝❡♥t❛❣❡♠✳ P❛r❛ ❛ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ ❞❡ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❡①❡♠♣❧♦s ✈❡❥❛ ♦ ❆♣ê♥❞✐❝❡✳
✶✳✶ Pr♦❣r❡ssõ❡s ❆r✐t♠ét✐❝❛s
❯♠ t❡❛tr♦ ♣♦ss✉✐12♣♦❧tr♦♥❛s ♥❛ ♣r✐♠❡✐r❛ ✜❧❡✐r❛✱14♥❛ s❡❣✉♥❞❛✱ 16♥❛ t❡r❝❡✐r❛❀
❛s ❞❡♠❛✐s ✜❧❡✐r❛s s❡ ❝♦♠♣õ❡ ♥❛ ♠❡s♠♦ s❡q✉ê♥❝✐❛✳ ❈♦♠♦ s❛❜❡r q✉❛♥t❛s ✜❧❡✐r❛s sã♦ ♥❡❝❡ssár✐❛s ♣❛r❛ q✉❡ ♦ t❡❛tr♦ ♣♦ss✉❛ ✉♠ t♦t❛❧ ❞❡ 620 ♣♦❧tr♦♥❛s❄ ◆♦t❡ q✉❡
♦ ❝r❡s❝✐♠❡♥t♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❝❛❞❡✐r❛s ❞♦ t❡❛tr♦ ❝r❡s❝❡ ❞❡ ❢♦r♠❛ ✉♥✐❢♦r♠❡✳ ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ❡st❡ ♥ú♠❡r♦ ❝r❡s❝❡ ❞❡2❡♠ 2✳ ❊st❡ ♥ú♠❡r♦✱ ❝♦♠♦ ✈❡r❡♠♦s ❛❞✐❛♥t❡✱
é ❝❤❛♠❛❞♦ ❞❡ r❛③ã♦ ❞❡st❛ s❡q✉ê♥❝✐❛ ❡ t❛❧ s❡q✉ê♥❝✐❛ é ❝❤❛♠❛❞❛ ❞❡ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛✱ ✈❡❥❛ ❬✶✷❪✳
❉❡✜♥✐çã♦ ✶ ❯♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ ✭P❆✮ é ✉♠❛ s❡q✉ê♥❝✐❛ ♥❛ q✉❛❧ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❝❛❞❛ t❡r♠♦ ❡ ♦ t❡r♠♦ ❛♥t❡r✐♦r é ❝♦♥st❛♥t❡✳ ❊ss❛ ❞✐❢❡r❡♥ç❛ é ❝❤❛♠❛❞❛
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✶✳ Pr♦❣r❡ssõ❡s ❆r✐t♠ét✐❝❛s
r❛③ã♦ ❞❛ ♣r♦❣r❡ssã♦ ❡ é r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❧❡tr❛ r✳
❯♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ s❡rá r❡♣r❡s❡♥t❛❞❛ ♣♦r (a1, a2, a3, . . . , an, . . .)✳ P❛r❛
❛✈❛♥ç❛r ❞♦ ♣r✐♠❡✐r♦ t❡r♠♦ ♣❛r❛ ♦ s❡❣✉♥❞♦✱ ❜❛st❛ s♦♠❛r r ❛ ❡st❡ t❡r♠♦✱ ♦✉ s❡❥❛✱ a2 =a1+r✳ ❏á ♣❛r❛ ❛✈❛♥ç❛r ❞♦ s❡❣✉♥❞♦ ♣❛r❛ ♦ t❡r❝❡✐r♦ t❡r♠♦✱ ❢❛③❡♠♦s ♥♦✈❛♠❡♥t❡ a3 = a2 +r✳ ❊♠ ❣❡r❛❧✱ ✈❛♠♦s ❝♦♥s❡❣✉✐r ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦ n✲és✐♠♦ t❡r♠♦ ❡♠
❢✉♥çã♦ ❞♦ t❡r♠♦ ❛♥t❡r✐♦r✿
an=an−1+r.
❆ss✐♠✱ t❡♠♦s ♦ s❡❣✉✐♥t❡
r =a2−a1,
r =a3−a2,
✳✳✳
r=an−1−an−2,
r=an−an−1.
P♦rt❛♥t♦✱ s♦♠❛♥❞♦ ❡st❛s n−1 ❡q✉❛çõ❡s✱ ♦❜t❡♠♦s q✉❡
an =a1+ (n−1)r. ✭✶✳✶✮
❊st❛ ❡①♣r❡ssã♦ é ❝❤❛♠❛❞❛ ❞❡ t❡r♠♦ ❣❡r❛❧ ❞❛ P❆✳
◆✉♠❛ s✐t✉❛çã♦ ❡♠ q✉❡ ❤á ❡♠♣rést✐♠♦ ❞❡ ❞✐♥❤❡✐r♦ ♣❛r❛ ❞❡✈♦❧✉çã♦ ❞❡♣♦✐s ❞❡ ✉♠ ❝❡rt♦ ♥ú♠❡r♦ ❞❡ ♣❡rí♦❞♦s✱ ❡ ❡♠ q✉❡ ❡ss❡ ❡♠♣rést✐♠♦ é ❜❛s❡❛❞♦ ♥♦ s✐st❡♠❛ ❞❡ ❥✉r♦s s✐♠♣❧❡s✱ ♦s ❥✉r♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛ ❝❛❞❛ ♣❡rí♦❞♦ sã♦ ❝♦♥st❛♥t❡s ❡ ✐❣✉❛✐s ❛♦ ✈❛❧♦r ❝❛❧❝✉❧❛❞♦ ♥♦ ✜♠ ❞♦ ♣r✐♠❡✐r♦ ♣❡rí♦❞♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♥♦ ✜♠ ❞♦ ♣r✐♠❡✐r♦ ♣❡rí♦❞♦✱ ♦s ❥✉r♦s sã♦ ❛❝r❡s❝✐❞♦s ❛♦ ❝❛♣✐t❛❧ ✐♥✐❝✐❛❧✱ r❡s✉❧t❛♥❞♦ ♥♦ ♠♦♥t❛♥t❡ M1✳ ◆♦
✜♠ ❞♦ s❡❣✉♥❞♦ ♣❡rí♦❞♦✱ ♦s ❥✉r♦s sã♦ ❛❝r❡s❝✐❞♦s ❛♦ ♠♦♥t❛♥t❡ M1✱ r❡s✉❧t❛♥❞♦ ♥♦
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✶✳ Pr♦❣r❡ssõ❡s ❆r✐t♠ét✐❝❛s
♠♦♥t❛♥t❡ M2✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡ ❛té ♦ ✜♠ ❞♦s ♣❡rí♦❞♦s ❝♦♥tr❛t❛❞♦s✱ ❡♠ q✉❡ ♦
❝❛♣✐t❛❧ ❡♠♣r❡st❛❞♦ t❡rá s❡ tr❛♥s❢♦r♠❛❞♦ ♥♦ ♠♦♥t❛♥t❡ Mn✳
❱❛♠♦s ❝♦♥s✐❞❡r❛r✱ ❡♥tã♦✱ ✉♠ ❡♠♣rést✐♠♦ ❞❡ ♠✐❧ r❡❛✐s ❛ s❡r ♣❛❣♦ ❛♦ ✜♠ ❞❡ ♦✐t♦ ♠❡s❡s ❛ t❛①❛ ❞❡ 2% ❛♦ ♠ês✱ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠ s✐st❡♠❛ ❞❡ ❥✉r♦s s✐♠♣❧❡s✳ ◗✉❛♥t♦
❞❡✈❡rá s❡r ♣❛❣♦ ♣❛r❛ ❛ q✉✐t❛çã♦ ❞❛ ❞í✈✐❞❛❄ ◆♦t❡ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ 2% ❛♦ ♠ês
❞❡ ♠✐❧ r❡❛✐s sã♦ ✷✵ r❡❛✐s✳ ■st♦ ❢❛③ ❝♦♠ q✉❡ t❡♥❤❛♠♦s ❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s ♣❛r❛ ♦s ♠♦♥t❛♥t❡s✿
M1 = 1000 + 20,
M2 = 1020 + 20,
M3 = 1040 + 20,
✳✳✳
❆ss✐♠✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ❛ ❢ór♠✉❧❛ ❞♦ t❡r♠♦ ❣❡r❛❧ ♣❛r❛ ✉♠❛ P❆ ♣❛r❛ ❝❛❧❝✉❧❛r ♦ ❞✐♥❤❡✐r♦ ❛ s❡r ♣❛❣♦ ❛♦ ✜♥❛❧ ❞♦s ♦✐t♦ ♠❡s❡s✱ ♦✉ s❡❥❛✱ t❡♠♦s q✉❡
M8 =M1+ (8−1)·20 = 1020 + 7·20 = 1160.
❑❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ss ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦ q✉❡ ✈✐✈❡✉ ❡♥tr❡ 1777 ❛ 1855✳
❈❡rt♦ ❞✐❛✱ q✉❛♥❞♦ ❡❧❡ ❡r❛ ✉♠ ❡st✉❞❛♥t❡ ❝♦♠ ♠❡♥♦s ❞❡10❛♥♦s ❞❡ ✐❞❛❞❡✱ s❡✉ ♣r♦❢❡ss♦r✱
q✉❡r❡♥❞♦ ♠❛♥t❡r ♦ s✐❧ê♥❝✐♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ♣♦r ✉♠ ❜♦♠ t❡♠♣♦✱ ♣❡❞✐✉ q✉❡ ♦s ❛❧✉♥♦s s♦♠❛ss❡♠ t♦❞♦s ♦s ♥ú♠❡r♦s ❞❡ 1 ❛ 100✱ ✐st♦ é✱ 1 + 2 + 3 + 4 + . . .+ 99 + 100✳
P❛r❛ ❛ s✉r♣r❡s❛ ❞♦ ♣r♦❢❡ss♦r✱ ❞❡♣♦✐s ❞❡ ❛❧❣✉♥s ♠✐♥✉t♦s ●❛✉ss ❞✐ss❡ q✉❡ ❛ s♦♠❛ ❡r❛ 5050✳ ❊st❡ ❢❛t♦ é ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ú❜❧✐❝♦ ❡ ♣♦❞❡ s❡r ✈✐st♦ ❡♠ ❬✻❪✳ ◆♦t❡
q✉❡ ●❛✉ss s♦♠♦✉ ♥❛❞❛ ♠❛✐s q✉❡ ♦s ✶✵✵ ♣r✐♠❡✐r♦s t❡r♠♦s ❞❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛
1, 2, · · ·, n, · · · ❞❡ r❛③ã♦ ❡ ❞❡ ♣r✐♠❡✐r♦ t❡r♠♦ ✐❣✉❛✐s ❛ 1✳ ❱❛♠♦s ❞❡s❝r❡✈❡r ❛ s❡❣✉✐r
❝♦♠♦ ♦❜t❡r ❡st❛ s♦♠❛ ♣❛r❛ ✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ q✉❛❧q✉❡r✳ ❙❡❥❛(a1, a2, a3, . . .)
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✶✳ Pr♦❣r❡ssõ❡s ❆r✐t♠ét✐❝❛s
✉♠❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ ❞❡ r❛③ã♦ r✳ ❙❡❥❛ t❛♠❜é♠ Sn = a1 +a2 +a3 +. . .+an
❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❞❛ P❆✱ q✉❡ t❛♠❜é♠ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡
❢♦r♠❛✿ Sn =an+an−1+. . .+a3+a2+a1✳ ❙♦♠❛♥❞♦ ❡st❛s ❞✉❛s ❡①♣r❡ssõ❡s✱ t❡♠♦s
q✉❡
2Sn = (a1+an) + (a2+an−1) +. . .+ (an−1+a2) + (an+a1).
❆♦ ❛♥❛❧✐s❛r ❝❛❞❛ ❡①♣r❡ssã♦ ❡♥tr❡ ♣❛rê♥t❡s❡s ❞❛ s♦♠❛ ❛❝✐♠❛✱ ✈❡r✐✜❝❛♠♦s q✉❡ t♦❞♦s ❡❧❡s tê♠ ♦ ♠❡s♠♦ ✈❛❧♦r ❡ ✐❣✉❛❧ a1+an ♣♦✐s
aj +an−(j−1) = a1+ (j −1)r+a1+ (n−(j−1)−1)r
= a1+ [a1+ (n+ (j−1)−(j−1)−1)r]
= a1+ [a1+ (n−1)r]
= a1+an
♣❛r❛ ❝❛❞❛ j = 2, . . . , n−1✳ P♦rt❛♥t♦✱ 2Sn=n(a1+an)✱ ♦✉ s❡❥❛✱
Sn =
(a1+an)n
2 . ✭✶✳✷✮
❱❡❥❛ ✉♠ ❡①❡♠♣❧♦ ♦♥❞❡ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ❡st❛ ❢ór♠✉❧❛✳
❊①❡♠♣❧♦ ✶ ❯♠ ❝✐❝❧✐st❛ ♣❡r❝♦rr❡ 20 q✉✐❧ô♠❡tr♦s ♥❛ ♣r✐♠❡✐r❛ ❤♦r❛✱ 17 q✉✐❧ô♠❡tr♦s
♥❛ s❡❣✉♥❞❛ ❤♦r❛✱ 14 ♥❛ t❡r❝❡✐r❛✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ ❡♠ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛✳
❱❛♠♦s ❝❛❧❝✉❧❛r q✉❛♥t♦s q✉✐❧ô♠❡tr♦s ♦ ❝✐❝❧✐st❛ ♣❡r❝♦rr❡✉ ❡♠ 5 ❤♦r❛s✳ P❛r❛ ✐ss♦✱
♥♦t❡ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ ❛ r❛③ã♦ ❞❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ é ❞❡ −3 q✉✐❧ô♠❡tr♦s ❡
q✉❡ ♦ ♣r✐♠❡✐r♦ t❡r♠♦ ❞❛ s❡q✉ê♥❝✐❛ é 20✳ ❈♦♠♦ q✉❡r❡♠♦s ❝❛❧❝✉❧❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡
q✉✐❧ô♠❡tr♦s q✉❡ ♦ ❝✐❝❧✐st❛ ♣❡r❝♦rr❡✉✱ é ♥❡❝❡ssár✐♦ ✉t✐❧✐③❛r ❛ ❢ór♠✉❧❛ ❞♦ t❡r♠♦ ❣❡r❛❧ ♣❛r❛ ❝❛❧❝✉❧❛r ♦ q✉✐♥t♦ t❡r♠♦ ❞❛ s❡q✉ê♥❝✐❛ ❡ ❡♠ s❡❣✉✐❞❛ ❛ ❢ór♠✉❧❛ ❞❛ s♦♠❛ ❞♦s ❝✐♥❝♦
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✷✳ Pr♦❣r❡ssõ❡s ●❡♦♠étr✐❝❛s
♣r✐♠❡✐r♦s t❡r♠♦s ❞❛ P❆✳ ❚❡♠♦s q✉❡
a5 =a1+ (5−1)r= 20 + (5−1)·(−3) = 8
❡✱ ♣♦rt❛♥t♦✱
S5 =
(a1+a5)·5
2 =
(20 + 8)·5 2 = 70.
✶✳✷ Pr♦❣r❡ssõ❡s ●❡♦♠étr✐❝❛s
❯♠❛ ❜♦❧❛ ❞❡ ❜♦rr❛❝❤❛ ❝❛✐ ❞❡ ✉♠❛ ❛❧t✉r❛h✳ ❆♣ós ❝❤♦❝❛r✲s❡ ❝♦♠ ♦ s♦❧♦✱ ❛t✐♥❣❡
✉♠❛ ❛❧t✉r❛ ✐❣✉❛❧ ❛ 2/3 ❞❛ ❛❧t✉r❛ ❛♥t❡r✐♦r ❡ ❡st❛ ♠❡s♠❛ s✐t✉❛çã♦ s❡ ♠❛♥té♠ ♥♦s
❝❤♦q✉❡s s✉❜s❡q✉❡♥t❡s ❛té q✉❡ ❛ ❜♦❧❛ ♣❛r❛✳ ❈♦♠♦ ♣♦❞❡r❡♠♦s ❝❛❧❝✉❧❛r ❛ ❞✐stâ♥❝✐❛ ♣❡r❝♦rr✐❞❛ ♣❡❧❛ ❜♦❧❛ ❛té q✉❡ ❡❧❛ ♣❛r❡❄ ◆♦t❡ q✉❡ ❛ ❛❧t✉r❛ ❞❛ ❜♦❧❛ s❡♠♣r❡ ❞✐♠✐♥✉✐
1/3❞❛ ❛❧t✉r❛ ❛♥t❡r✐♦r✳ ❊st❡ ♥ú♠❡r♦ é ❝❤❛♠❛❞♦ ❞❡ r❛③ã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❛❧t✉r❛s ❡
t❛❧ s❡q✉ê♥❝✐❛ é ❞❡♥♦♠✐♥❛❞❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛✳
❉❡✜♥✐çã♦ ✷ ❯♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ✭P●✮ é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♥ã♦ ♥✉❧♦s ♥❛ q✉❛❧ é ❝♦♥st❛♥t❡ ♦ q✉♦❝✐❡♥t❡ ❞❛ ❞✐✈✐sã♦ ❞❡ ❝❛❞❛ t❡r♠♦ ♣❡❧♦ t❡r♠♦ ❛♥t❡r✐♦r✳ ❊st❡ q✉♦❝✐❡♥t❡ é ❝❤❛♠❛❞♦ ❞❡ r❛③ã♦ ❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r q✱ ✈❡❥❛ ❬✶✷❪✳
❯♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ s❡rá r❡♣r❡s❡♥t❛❞❛ ♣♦r (a1, a2, a3, . . . , an, . . .)✳ P❛r❛
❛✈❛♥ç❛r ❞♦ ♣r✐♠❡✐r♦ t❡r♠♦ ♣❛r❛ ♦ s❡❣✉♥❞♦✱ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛rq❛ ❡st❡ t❡r♠♦✱ ♦✉ s❡❥❛✱ a2 = a1·q✳ ❏á ♣❛r❛ ❛✈❛♥ç❛r ❞♦ s❡❣✉♥❞♦ ♣❛r❛ ♦ t❡r❝❡✐r♦ t❡r♠♦✱ ❢❛③❡♠♦s ♥♦✈❛♠❡♥t❡ a3 =a2·q✳ ❊♠ ❣❡r❛❧✱ ✈❛♠♦s ❝♦♥s❡❣✉✐r ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦ n✲t❡r♠♦ ❡♠ ❢✉♥çã♦ ❞♦
t❡r♠♦ ❛♥t❡r✐♦r✿
an=an−1·q.
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✷✳ Pr♦❣r❡ssõ❡s ●❡♦♠étr✐❝❛s
❙✉❜st✐t✉✐♥❞♦ ❛ ❡①♣r❡ssã♦ ❞♦ t❡r♠♦ s❡❣✉✐♥t❡ ♣❡❧♦ ❛♥t❡r✐♦r✱ ❝♦♥s❡❣✉✐♠♦s ❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦ t❡r♠♦ ❣❡r❛❧ ❞❡ ✉♠❛ P● ❞❛❞♦ ♣♦r
an =a1 ·qn−1. ✭✶✳✸✮
P♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠❛ s✐t✉❛çã♦ s❡♠❡❧❤❛♥t❡ ❛ q✉❡ tr❛t❛♠♦s ♥♦ ❝❛s♦ ❞❡ ✉♠❛ P❆✱ ♦✉ s❡❥❛✱ ❡♠ q✉❡ ❤á ❡♠♣rést✐♠♦ ❞❡ ❞✐♥❤❡✐r♦ ♣❛r❛ ❞❡✈♦❧✉çã♦ ❡♠ ✉♠ ❝❡rt♦ ♥ú♠❡r♦ ❞❡ ♣❡rí♦❞♦s✱ ♠❛s ❡♠ q✉❡ ♦ ❡♠♣rést✐♠♦ é ❜❛s❡❛❞♦ ♥♦ s✐st❡♠❛ ❞❡ ❥✉r♦s ❝♦♠♣♦st♦s✳ ❖s ❥✉r♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛ ❝❛❞❛ ♣❡rí♦❞♦ ❞❡st❛ ✈❡③ ♥ã♦ sã♦ ❝♦♥st❛♥t❡s ❡ ♣♦r ✐ss♦ ♣r❡❝✐s❛♠ s❡r ❝❛❧❝✉❧❛❞♦s ❛♦ ✜♠ ❞❡ ❝❛❞❛ ♣❡rí♦❞♦ r❡❧❛t✐✈♦ ❛♦ ♠♦♥t❛♥t❡ ❛t✉❛❧ ❞❛ ❞í✈✐❞❛✳ ❉❡ss❛ ❢♦r♠❛✱ ♥♦ ✜♠ ❞♦ ♣r✐♠❡✐r♦ ♣❡rí♦❞♦✱ ♦s ❥✉r♦s sã♦ ❛❝r❡s❝✐❞♦s ❛♦ ❝❛♣✐t❛❧ ✐♥✐❝✐❛❧✱ r❡s✉❧t❛♥❞♦ ♥♦ ♠♦♥t❛♥t❡ M1✳ ◆♦ ✜♠ ❞♦ s❡❣✉♥❞♦ ♣❡rí♦❞♦✱ ♦s ❥✉r♦s sã♦ r❡❝❛❧❝✉❧❛❞♦s
s♦❜r❡ ♦ ♠♦♥t❛♥t❡M1 ❡ s♦♠❛❞♦s✱ r❡s✉❧t❛♥❞♦ ♥♦ ♠♦♥t❛♥t❡M2❡✱ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ ❛té
♦ ✜♠ ❞♦s ♣❡rí♦❞♦s ❝♦♥tr❛t❛❞♦s✱ ❡♠ q✉❡ ♦ ❝❛♣✐t❛❧ ❡♠♣r❡st❛❞♦ t❡rá s❡ tr❛♥s❢♦r♠❛❞♦ ♥♦ ♠♦♥t❛♥t❡ Mn✳
❱❛♠♦s ❝♦♥s✐❞❡r❛r ♥♦✈❛♠❡♥t❡ ✉♠ ❡♠♣rést✐♠♦ ❞❡ ♠✐❧ ❞❡ r❡❛✐s ❛ s❡r❡♠ ♣❛❣♦s ❛♦ ✜♠ ❞❡ ♦✐t♦ ♠❡s❡s ❛ t❛①❛ ❞❡ 2% ❛♦ ♠ês✱ s❡♥❞♦ q✉❡ ❞❡st❛ ✈❡③ ♦ s✐st❡♠❛ ❝♦♥s✐❞❡r❛❞♦
é ♦ ❞❡ ❥✉r♦s ❝♦♠♣♦st♦s✳ ◗✉❛♥t♦ ❞❡✈❡rá s❡r ♣❛❣♦ ♣❛r❛ ❛ q✉✐t❛çã♦ ❞❛ ❞í✈✐❞❛❄ ◆♦t❡ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡2%❛♦ ♠ês ❞❡ 1.000 r❡❛✐s sã♦20r❡❛✐s ❡ q✉❡2%❞❡1.020r❡❛✐s sã♦ 20 r❡❛✐s ❡ 40 ❝❡♥t❛✈♦s✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ■st♦ ❢❛③ ❝♦♠ q✉❡ t❡♥❤❛♠♦s ❛s s❡❣✉✐♥t❡s
❡q✉❛çõ❡s ♣❛r❛ ♦s ♠♦♥t❛♥t❡s✿
M1 = 1000·1,02 = 1020,00
M2 = 1020·1,02 = 1040,00,
M3 = 1040,40·1,02≈1.061,21,
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✷✳ Pr♦❣r❡ssõ❡s ●❡♦♠étr✐❝❛s
❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ P♦❞❡♠♦s ✉t✐❧✐③❛r ❛ ❢ór♠✉❧❛ ❞♦ t❡r♠♦ ❣❡r❛❧ ♣❛r❛ ✉♠❛ P● ♣❛r❛ ❝❛❧❝✉❧❛r ♦ ❞✐♥❤❡✐r♦ ❛ s❡r ♣❛❣♦ ❛♦ ✜♥❛❧ ❞♦s ♦✐t♦ ♠❡s❡s✱ ♦✉ s❡❥❛✱ t❡♠♦s q✉❡
M8 =M1·q8−1 = 1020·(1,02)7 = 1.171,66
r❡❛✐s✳ ❈♦♠♦ ✜③❡♠♦s ♥♦ ❝❛s♦ ❞❛s ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s✱ t❛♠❜é♠ é ✐♥t❡r❡ss❛♥t❡ s❛❜❡r♠♦s ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ✉♠❛ P●✳ ➱ ♦ q✉❡ ❢❛r❡♠♦s ❛
♣❛rt✐r ❞❡ ❛❣♦r❛✳ ❙❡❥❛
Sn =a1+a1·q+a1·q2 +. . .+a1·qn−1
❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ♥ã♦ ❝♦♥st❛♥t❡ ✭♦✉
s❡❥❛✱ q6= 1✮✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ Sn ♣♦r q✱ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿
q·Sn =a1 ·q+a1·q2+. . .+a1 ·qn
❡ s✉❜tr❛✐♥❞♦ ❞❡ Sn✱ t❡♠♦s q✉❡
Sn−q·Sn =a1 −a1·qn,
♦✉ s❡❥❛✱
Sn =
a1·(qn−1)
q−1 , ✭✶✳✹✮
❝♦♠ q 6= 1✳ ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦✳
❊①❡♠♣❧♦ ✷ ❯♠❛ ♣❡ss♦❛ ❛♣♦st❛ ♥❛ ❧♦t❡r✐❛ ❞✉r❛♥t❡ ✶✵ s❡♠❛♥❛s✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❡♠ ❝❛❞❛ s❡♠❛♥❛ ♦ ✈❛❧♦r ❞❛ ❛♣♦st❛ é ♦ ❞♦❜r♦ ❞♦ ✈❛❧♦r ❞❛ ❛♣♦st❛ ❞❛ s❡♠❛♥❛ ❛♥t❡r✐♦r✳ ❙❡ ♦ ✈❛❧♦r ❞❛ ❛♣♦st❛ ❞❛ ♣r✐♠❡✐r❛ s❡♠❛♥❛ é ✶✵ r❡❛✐s✱ ✈❛♠♦s ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r t♦t❛❧ ❛♣♦st❛❞♦ ❛♦ ✜♥❛❧ ❞❡ ✶✵ s❡♠❛♥❛s✳ ◆♦t❡ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ ❡st❛♠♦s tr❛t❛♥❞♦ ❞❡ ✉♠❛ P● ❞❡ ♣r✐♠❡✐r♦ t❡r♠♦ ✶✵ ❡ r❛③ã♦ 2✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣♦❞❡♠♦s ✉s❛r ❛ ❢ór♠✉❧❛ ✭✶✳✹✮
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✸✳ ❘❛③ã♦ ❡ Pr♦♣♦rçã♦
♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛✳ ❚❡♠♦s q✉❡ ❝❛❧❝✉❧❛r S10✿
S10 = 10·(2
10−1)
2−1 = 10.230,00.
❆ss✐♠ ✈❡♠♦s q✉❡ ♥♦ ✜♥❛❧ ❞❡ ✶✵ s❡♠❛♥❛s ❢♦r❛♠ ❛♣♦st❛❞♦s R$10.230,00✳
❱❡❥❛♠♦s ❛❣♦r❛ ✉♠ ❡①❡♠♣❧♦ ❞❡ s♦♠❛s ✐♥✜♥✐t❛s✳ ❈♦♠♦ ❢❛r❡♠♦s ♣❛r❛ ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s t❡r♠♦s ❞❛ P●
1 2, 1 4, 1 8, 1 16,· · · ,
1 2n,· · ·
, ❝♦♠ n∈N?
❚❡♠♦s
Sn =
a1·(qn−1)
q−1 ,
❝♦♠ q 6= 1✳ ❯♠❛ ✈❡③ q✉❡ 0 < q = 1/2 < 1✱ ✈❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ (qn) ❝♦♥✈❡r❣❡
♣❛r❛ ③❡r♦ q✉❛♥❞♦nt❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✱ ❝♦♠♦ ✈✐♠♦s ♥♦ ❊①❡♠♣❧♦ ✸✻ ❞♦ ❆♣ê♥❞✐❝❡✳ ❉❛í✱
t❡♠♦s q✉❡
lim
n→∞Sn=
a1
1−q,
❡✱ ♣♦rt❛♥t♦✱ 1 2+ 1 4+ 1
8+. . .+ 1
2n +. . .=
1/2
1−1/2 = 1.
✶✳✸ ❘❛③ã♦ ❡ Pr♦♣♦rçã♦
Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ é ✉♠ ❞♦s ❝♦♥t❡ú❞♦s ♠❛✐s ✉t✐❧✐③❛❞♦s ♥♦ ♥♦ss♦ ❞✐❛ ❛ ❞✐❛✳ ❊st❛♠♦s ❝♦♥st❛♥t❡♠❡♥t❡ ❝♦♠♣❛r❛♥❞♦ ❣r❛♥❞❡③❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ ♠❛ss❛s✱ ✈❡❧♦❝✐❞❛❞❡s✱ t❡♠♣♦✱ ❢♦r♠❛s✱ t❛♠❛♥❤♦s✱ ❡t❝✳ ❊♥✜♠✱ t✉❞♦ ♦ q✉❡ ♥♦s ❝❡r❝❛✳ ❊ss❛s ❝♦♠♣❛r❛çõ❡s ♠✉✐t❛s ✈❡③❡s ❢❛❝✐❧✐t❛♠ ♥❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s✳
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✸✳ ❘❛③ã♦ ❡ Pr♦♣♦rçã♦
✶✳✸✳✶ ❘❛③ã♦
❘❛③ã♦ é ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❞✉❛s ❣r❛♥❞❡③❛s ❞♦ ♠❡s♠♦ t✐♣♦✳ ❊①♣r❡ss❛♠♦s ❣❡r❛❧♠❡♥t❡ ♥❛ ❢♦r♠❛ ✏a ♣❛r❛ b✑✱ ♦♥❞❡ a ❡ b sã♦ ♥ú♠❡r♦s✱ ♦✉ ❛✐♥❞❛ ❛❧❣✉♠❛s ✈❡③❡s
r❡♣r❡s❡♥t❛❞❛ ❝♦♠♦ ✉♠ q✉♦❝✐❡♥t❡ ❞❡ ❞✉❛s q✉❛♥t✐❞❛❞❡s q✉❡ ✐♥❞✐❝❛♠ ❡①♣❧✐❝✐t❛♠❡♥t❡ q✉❛♥t❛s ✈❡③❡s ♦ ♣r✐♠❡✐r♦ ♥ú♠❡r♦ ❝♦♥té♠ ♦ s❡❣✉♥❞♦ ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✮✳ ❉❡♥♦t❛r❡♠♦s ❛ r❛③ã♦ ❡♥tr❡ a ❡ b ♣♦r a/b✱ ♦♥❞❡ b6= 0✳
❊①❡♠♣❧♦ ✸ ❆♥❞ré ❢❛③ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞✐ár✐♦ ❞❡ 1,4 q✉✐❧ô♠❡tr♦s ❞❡ ❝❛s❛ ♣❛r❛ ❛
❯❋P❇✱ ♦♥❞❡ ❢❛③ ♠❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛✳ ❊st❡ ❞❡s❧♦❝❛♠❡♥t♦ é ♣❡r❝♦rr✐❞♦ ♣♦r ❡❧❡ ❡♠ 10♠✐♥✉t♦s✳ ❱❛♠♦s ❝❛❧❝✉❧❛r ❛ r❛③ã♦ ❡♥tr❡ ❛ ❞✐stâ♥❝✐❛ ♣❡r❝♦rr✐❞❛ ❡ ♦ t❡♠♣♦ ❣❛st♦
♣❛r❛ ♣❡r❝♦rrê✲❧❛✿
1,4 ❦♠
10 ♠✐♥✉t♦s = 0,14 ❦♠✴♠✐♥✉t♦s.
P♦❞❡♠♦s ❛✐♥❞❛ ❞❡♥♦t❛r ❡ss❛ r❛③ã♦ ❝♦♠♦ s❡♥❞♦ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛✱ ✐st♦ é✱ ❛ ✈❡❧♦❝✐❞❛❞❡ q✉❡ ❆♥❞ré ❝❛♠✐♥❤❛ é ❞❡ 0,14 ❦♠✴♠✐♥✉t♦s✱ q✉❡✱ tr❛♥s❢♦r♠❛♥❞♦ ♦ t❡♠♣♦ ❡♠ ❤♦r❛s✱
♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r 8,4km/h✳
✶✳✸✳✷ Pr♦♣♦rçã♦
➱ ♠✉✐t♦ ❝♦♠✉♠ ♦❜s❡r✈❛r♠♦s ✉♠ ♣r♦❥❡t♦ ❛rq✉✐t❡tô♥✐❝♦ ♦✉ ✉♠❛ ✐♠❛❣❡♠ ❡ ❞✐③❡r♠♦s q✉❡ ✉♠❛ ❞❡ s✉❛s ♣❛rt❡s é ♠✉✐t♦ ♣❡q✉❡♥❛ ❡♠ r❡❧❛çã♦ ❛ ♦✉tr❛✱ ❝♦♠♦ ♦ q✉❛❞r♦ ❞❛ ❋✐❣✉r❛ ✶✳✶ ❝❤❛♠❛❞♦ ❆❜❛♣♦r✉ ❞❛ ❛rt✐st❛ ♣❧ást✐❝❛ ❚❛rs✐❧❛ ❞♦ ❆♠❛r❛❧✱ ♦❜s❡r✈❛♥❞♦ q✉❡ s✉❛s ♠❡❞✐❞❛s ♥ã♦ sã♦ ♣r♦♣♦r❝✐♦♥❛✐s ❡♥tr❡ s✐✳
❊ss❛ ❞❡s♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ✭✐♥t❡♥❝✐♦♥❛❧ ♦✉ ♥ã♦✮ é ♣❡r❝❡❜✐❞❛ q✉❛♥❞♦✱ ✐♥st✐♥t✐✈❛♠❡♥t❡✱ ❝♦♠♣❛r❛♠♦s ❛s ♠❡❞✐❞❛s ❞❡ss❛ ✐♠❛❣❡♠ ❝♦♠ ❛s ❞❡ ♦✉tr❛ q✉❡ t❡♠♦s ❝♦♠♦ ♣❛❞rã♦✳
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✸✳ ❘❛③ã♦ ❡ Pr♦♣♦rçã♦
❋✐❣✉r❛ ✶✳✶✿ ❆❜❛♣♦r✉✳ ❋♦♥t❡✿ ❛rt❡❞❡s❝r✐t❛✳❜❧♦❣s♣♦t✳❝♦♠✳❜r
❱♦❧t❡♠♦s ❛♦ ♥♦ss♦ ❝♦♥t❡①t♦ s♦❜r❡ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛✳ ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦✳
❊①❡♠♣❧♦ ✹ ❊♠ ❞✉❛s ✜❧✐❛✐s X ❡ Y ❞❡ ✉♠❛ ♠❡s♠❛ ✐♥st✐t✉✐çã♦ ✜♥❛♥❝❡✐r❛✱ ♥♦s
s❡r✈✐ç♦s ❞❡ ❘❍ ✭r❡❝✉rs♦s ❤✉♠❛♥♦s✮✱ ❢♦✐ ✈❡r✐✜❝❛❞❛ ❛ s❡❣✉✐♥t❡ s✐t✉❛çã♦✿ ❡♠ X ❤á 20
❢✉♥❝✐♦♥ár✐♦s ❞♦s q✉❛✐s 12 tê♠ ❝✉rs♦ s✉♣❡r✐♦r ❝♦♠♣❧❡t♦ ❡ ♦s ❞❡♠❛✐s ❝✉rs♦s té❝♥✐❝♦s✱
♦✉ ♥í✈❡❧ ♠é❞✐♦✱ ❡ Y ♣♦ss✉✐ 30 ❢✉♥❝✐♦♥ár✐♦s✱ ❞♦s q✉❛✐s 18 ♣♦ss✉❡♠ ❝✉rs♦ s✉♣❡r✐♦r
❝♦♠♣❧❡t♦✳ ❱❡r✐✜❝❛✲s❡ ❡♥tã♦ q✉❡ ❛ r❛③ã♦ ❡♥tr❡ ♦s ❢✉♥❝✐♦♥ár✐♦s q✉❡ ❛♣r❡s❡♥t❛♠ ❝✉rs♦ s✉♣❡r✐♦r ❝♦♠♣❧❡t♦ ❡ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❘❍ ❞❡ ❝❛❞❛ ✜❧✐❛❧ é✿
✜❧✐❛❧ X : 12
20 = 0,6,
✜❧✐❛❧ Y : 18
30 = 0,6.
❆ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❞✉❛s r❛③õ❡s r❡❝❡❜❡ ✉♠ ♥♦♠❡ ❡s♣❡❝✐❛❧✳ ❉✐③❡♠♦s q✉❡✱ ♥❡ss❛ ♦r❞❡♠✱ ♦s ♥ú♠❡r♦s 12,20,18 ❡ 30 ❢♦r♠❛♠ ✉♠❛ ♣r♦♣♦rçã♦✳ ❉❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱
❞❛❞♦s q✉❛tr♦ ♥ú♠❡r♦s r❡❛✐s ❞✐❢❡r❡♥t❡s ❞❡ ③❡r♦ a, b, c ❡ d✱ ❡♠ ✉♠❛ ❞❛❞❛ ♦r❞❡♠✱ s❡
❛ r❛③ã♦ ❡♥tr❡ ♦s ❞♦✐s ♣r✐♠❡✐r♦s ❢♦r ✐❣✉❛❧ à r❛③ã♦ ❡♥tr❡ ♦s ❞♦✐s ú❧t✐♠♦s✱ ♦✉ s❡❥❛✱ s❡
a/b =c/d✱ ❞✐③❡♠♦s q✉❡ ♦s ♥ú♠❡r♦sa, b, c❡d✱ ♥❡st❛ ♦r❞❡♠✱ ❢♦r♠❛♠ ✉♠❛ ♣r♦♣♦rçã♦
✭✈❡❥❛ ❬✷❪✮✳
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✸✳ ❘❛③ã♦ ❡ Pr♦♣♦rçã♦
✶✳✸✳✸ ●r❛♥❞❡③❛s ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s
❉✉❛s ❣r❛♥❞❡③❛s x ❡ y sã♦ ❞✐t❛s ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s s❡ ❝r❡s❝❡♠ ✭♦✉
❞❡❝r❡s❝❡♠✮ ❥✉♥t❛s s❡♠♣r❡ ♠❡❞✐❛♥t❡ ✉♠ ❢❛t♦r ❝♦♠✉♠✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦❜s❡r✈❛✲ s❡ q✉❡✿ ❞♦❜r❛♥❞♦ ♦ ✈❛❧♦r ❞❡ ✉♠❛ ❞❛s ❣r❛♥❞❡③❛s✱ ♦ ✈❛❧♦r ❞❛ ♦✉tr❛ ❣r❛♥❞❡③❛ t❛♠❜é♠ ❞♦❜r❛❀ tr✐♣❧✐❝❛♥❞♦ ♦ ✈❛❧♦r ❞❡ ✉♠❛ ❞❛s ❣r❛♥❞❡③❛s✱ ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ✈❛❧♦r ❞❛ ♦✉tr❛ t❛♠❜é♠ tr✐♣❧✐❝❛✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❉❡ ♠❛♥❡✐r❛ ❣❡r❛❧✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♦ ✈❛❧♦r ❞❡ ✉♠❛ ❞❛s ❣r❛♥❞❡③❛s ♣♦r ❝❡rt❛ ❝♦♥st❛♥t❡k ∈R✱ ❛ ♦✉tr❛ t❡rá s❡✉ ✈❛❧♦r t❛♠❜é♠ ♠✉❧t✐♣❧✐❝❛❞♦ ♣❡❧❛ ♠❡s♠❛ ❝♦♥st❛♥t❡ k✳
❯♠❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛❞❛ ♥❡st❡ ❝❛s♦ é q✉❡ ♦s ✈❛❧♦r❡s ❞❡ ✉♠❛ ❞❛s ❣r❛♥❞❡③❛s ❡ ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❛❧♦r❡s ❞❛ ♦✉tr❛ ❣r❛♥❞❡③❛✱ ❡♠ ✈✐st❛ ❞❛ ❞❡✜♥✐çã♦✱ ❣✉❛r❞❛♠ s❡♠♣r❡ ❛ ♠❡s♠❛ r❛③ã♦✳ ❖✉ s❡❥❛✱ s❡ x1 ❡ x2 sã♦ ❞♦✐s ✈❛❧♦r❡s ❞✐st✐♥t♦s ❞❛
❣r❛♥❞❡③❛ x ❡ y1 ❡ y2 sã♦ ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❛❧♦r❡s ❞❡y✱ t❡♠✲s❡ q✉❡
y1 x1 =
y2 x2.
❊①❡♠♣❧♦ ✺ ◆✉♠❛ ✜r♠❛ ❡♠ q✉❡ tr❛❜❛❧❤❛♠ 36 ❢✉♥❝✐♦♥ár✐♦s✱ ❡①✐st❡♠ 21
❝♦♠♣✉t❛❞♦r❡s✳ ❆♣ós ✉♠❛ ❣r❛♥❞❡ ❛♠♣❧✐❛çã♦✱ ❛ ✜r♠❛ ♣❛ss♦✉ ❛ t❡r 60 ❢✉♥❝✐♦♥ár✐♦s✳
■r❡♠♦s ❝❛❧❝✉❧❛r ♦ ♥ú♠❡r♦ ❞❡ ❝♦♠♣✉t❛❞♦r❡s q✉❡ ❞❡✈❡rã♦ s❡r ❛❞q✉✐r✐❞♦s ♣❛r❛ q✉❡ s❡ ♠❛♥t❡♥❤❛ ❛ ♣r♦♣♦rçã♦ ❡♥tr❡ ♦s ❢✉♥❝✐♦♥ár✐♦s ❡ ♦s ❝♦♠♣✉t❛❞♦r❡s ❡①✐st❡♥t❡s ❛♥t❡s ❞❛ ❛♠♣❧✐❛çã♦✳ ❈♦♠♦ s❡ tr❛t❛ ❞❡ ❣r❛♥❞❡③❛s ❞✐r❡t❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s ❡ ❢♦r❛♠ ❝♦♥tr❛t❛❞♦s 24 ♥♦✈♦s ❢✉♥❝✐♦♥ár✐♦s✱ t❡♠♦s q✉❡✿
36 21 =
24
x ,
♦♥❞❡ x é ♦ ✈❛❧♦r q✉❡ s❡ q✉❡r ❞❡s❝♦❜r✐r✳ ❋❛③❡♥❞♦ ❛s ❝♦♥t❛s✱ t❡♠♦s q✉❡ x = 14✱ ♦✉
s❡❥❛✱ ❛ ✜r♠❛ t❡rá q✉❡ ❛❞q✉✐r✐r 14 ♥♦✈♦s ❝♦♠♣✉t❛❞♦r❡s✳
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✸✳ ❘❛③ã♦ ❡ Pr♦♣♦rçã♦
✶✳✸✳✹ ●r❛♥❞❡③❛s ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s
❉✐③❡♠♦s q✉❡ ❞✉❛s ❣r❛♥❞❡③❡sx ❡ y sã♦ ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s s❡✱ ❡♥tr❡
❡❧❛s✱ ♦❜s❡r✈❛✲s❡ ♦ s❡❣✉✐♥t❡ ❝♦♠♣♦rt❛♠❡♥t♦✿ ❞♦❜r❛♥❞♦✲s❡ ♦ ✈❛❧♦r ❞❡ x✱ ♦ ✈❛❧♦r ❞❡ y
s❡rá r❡❞✉③✐❞♦ ❛ s✉❛ ♠❡t❛❞❡❀ tr✐♣❧✐❝❛♥❞♦✲s❡ ♦ ✈❛❧♦r ❞❡ x✱ ♦ ✈❛❧♦r ❞❡y r❡❞✉③✐rá ❛ s✉❛
t❡rç❛ ♣❛rt❡✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❉❡ ❢♦r♠❛ ❣❡r❛❧✱ ♠✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ❛ ✉♠ ✈❛❧♦r ❞❛ ❣r❛♥❞❡③❛ x ♣♦r ✉♠❛ ❝❡rt❛ ❝♦♥st❛♥t❡ r❡❛❧ k✱ y t❡rá s❡✉ ✈❛❧♦r ❞✐✈✐❞✐❞♦ ♣❡❧❛ ♠❡s♠❛
❝♦♥st❛♥t❡ k✳
❯♠❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣♦rt❛♥t❡ ❞❛ ♣r♦♣♦rçã♦ ✐♥✈❡rs❛ é q✉❡ ♦s ✈❛❧♦r❡s ❞❡ ✉♠❛ ❞❛s ❣r❛♥❞❡③❛s ❡ ♦s ✐♥✈❡rs♦s ❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❛❧♦r❡s ❞❛ ♦✉tr❛ ❣r❛♥❞❡③❛✱ ❡♠ ✈✐st❛ ❞❛ ❞❡✜♥✐çã♦✱ ❣✉❛r❞❛♠ s❡♠♣r❡ ❛ ♠❡s♠❛ r❛③ã♦✳ ❙❡♥❞♦ x1 ❡ x2 ❞♦✐s ✈❛❧♦r❡s ❞✐st✐♥t♦s ❞❛
❣r❛♥❞❡③❛ x ❡ y1 ❡ y2 ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❛❧♦r❡s ❞❡ y✱ t❡♠✲s❡ q✉❡✿
y1
1
x1
= y2 1
x2 ,
♦✉ ❛✐♥❞❛✱
y1·x1 =y2·x2.
❊①❡♠♣❧♦ ✻ ❉♦✐s té❝♥✐❝♦s ❡♠ ❝♦♥t❛❜✐❧✐❞❛❞❡✱ ❲♦❧♥❡② ❡ ❘♦❣ér✐♦ q✉❡ ♣♦ss✉❡♠ 24❡ 36
❛♥♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✈ã♦ r❡♣❛rt✐r ❡♥tr❡ s✐ ✉♠ t♦t❛❧ ❞❡ 220 ♣r♦❝❡ss♦s ❞❡ ❛✉❞✐t♦r✐❛✱
♣❛r❛ ❝♦♥❢❡r✐r ♦s ❝á❧❝✉❧♦s✳ ❖ t♦t❛❧ ❞❡ ♣r♦❝❡ss♦s ❢♦✐ r❡♣❛rt✐❞♦ ❡♠ ♣❛rt❡s ✐♥✈❡rs❛♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✐s às s✉❛s r❡s♣❡❝t✐✈❛s ✐❞❛❞❡s✱ ♦✉ s❡❥❛✱ ✉s❛r❡♠♦s ❛ r❡❧❛çã♦ ❡♥❝♦♥tr❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♣r♦❝❡ss♦s q✉❡ ❝❛❜❡ ❛ ❝❛❞❛ ✉♠✳ ❈♦♠ ❡❢❡✐t♦✱ ♥♦t❡ q✉❡ s❡ ❲♦❧♥❡② ❛✉❢❡r✐r x ♣r♦❝❡ss♦s✱ ❡♥tã♦ ❘♦❣ér✐♦ ❛✉❢❡r✐rá ♦ r❡st❛♥t❡✱ ♦✉
s❡❥❛✱ 220−x ♣r♦❝❡ss♦s✳ ▲♦❣♦✱
x·24 = (220−x)·36
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✹✳ P♦r❝❡♥t❛❣❡♠
❡✱ ♣♦rt❛♥t♦✱
x= 126.
❈♦♥❝❧✉í♠♦s✱ ❡♥tã♦✱ q✉❡ ❝❛❜❡rá ❛ ❲♦❧♥❡② 126 ♣r♦❝❡ss♦s✱ ❡♥q✉❛♥t♦ ❛ ❘♦❣ér✐♦ 84✳
✶✳✹ P♦r❝❡♥t❛❣❡♠
❉♦ ✜♥❛♥❝✐❛♠❡♥t♦ ❞♦ ❝❛rr♦ às ♣r♦♠♦çõ❡s ❞❛s ❧♦❥❛s✱ q✉❛s❡ t✉❞♦ ♦ q✉❡ ❡♥✈♦❧✈❡ ❛s ❝♦♥t❛s ❞♦s ❜r❛s✐❧❡✐r♦s tr❛③ ♦ s✐♥❛❧ ❞❡ ✏♣♦r❝❡♥t❛❣❡♠✑✳ ❈♦♠ ❛ ❡①♣❛♥sã♦ ❞♦ ❝ré❞✐t♦ ❡ ❛ ♠❛✐♦r ♦❢❡rt❛ ❞❡ ✐♥✈❡st✐♠❡♥t♦s ♥♦s ú❧t✐♠♦s ❛♥♦s✱ ❡❧❡ ❛♣❛r❡❝❡ ❝❛❞❛ ✈❡③ ♠❛✐s ♥♦ ❥✉r♦ ❞♦ ❡♠♣rést✐♠♦✱ ♥❛ r❡♠✉♥❡r❛çã♦ ❞❛ ♣♦✉♣❛♥ç❛✱ ♥♦s ♣r❡ç♦s ❞❛s ❛çõ❡s✱ ❡t❝✳ ▼❛s ♠✉✐t❛ ❣❡♥t❡ ❛✐♥❞❛ t❡♠ ❞ú✈✐❞❛s s♦❜r❡ ❝♦♠♦ ❢❛③❡r t❛✐s ❝♦♥t❛s✳
❚r❛r❡♠♦s ❛q✉✐ ✉♠ ❜r❡✈❡ ❤✐stór✐❝♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ♣♦r❝❡♥t❛❣❡♥s q✉❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ♥❛ r❡❢❡rê♥❝✐❛ ❬✼❪✳ ✏❘❡❧❛t♦s ❤✐stór✐❝♦s ❞❛t❛♠ q✉❡ ♦ s✉r❣✐♠❡♥t♦ ❞♦s ❝á❧❝✉❧♦s ♣❡r❝❡♥t✉❛✐s ❛❝♦♥t❡❝❡✉ ♣♦r ✈♦❧t❛ ❞♦ sé❝✉❧♦ ■ ❛✳❈✳✱ ♥❛ ❝✐❞❛❞❡ ❞❡ ❘♦♠❛✳ ◆❡ss❡ ♣❡rí♦❞♦✱ ♦ ✐♠♣❡r❛❞♦r r♦♠❛♥♦ ❞❡❝r❡t♦✉ ✐♥ú♠❡r♦s ✐♠♣♦st♦s ❛ s❡r❡♠ ❝♦❜r❛❞♦s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♠❡r❝❛❞♦r✐❛ ♥❡❣♦❝✐❛❞❛✳ ❯♠ ❞♦s ✐♠♣♦st♦s ❝r✐❛❞♦s ♣❡❧♦s ❝❤❡❢❡s r♦♠❛♥♦s ❡r❛ ❞❡♥♦♠✐♥❛❞♦ ❝❡♥tés✐♠♦ r❡r✉♠ ✈❡♥❛❧✐✉♠✱ ❡ ♦❜r✐❣❛✈❛ ♦ ❝♦♠❡r❝✐❛♥t❡ ❛ ♣❛❣❛r ✉♠ ❝❡♥tés✐♠♦ ♣❡❧❛ ✈❡♥❞❛ ❞❛s ♠❡r❝❛❞♦r✐❛s ♥♦ ♠❡r❝❛❞♦✳ ◆❛q✉❡❧❛ é♣♦❝❛✱ ♦ ❝♦♠ér❝✐♦ ❞❡ ❡s❝r❛✈♦s ❡r❛ ✐♥t❡♥s♦ ❡ s♦❜r❡ ❛s ✈❡♥❞❛s ❡r❛ ❝♦❜r❛❞♦ ✉♠ ✐♠♣♦st♦ ❞❡1/25✭✉♠ ✈✐♥t❡ ❡
❝✐♥❝♦ ❛✈♦s✮✳
❖s ❝á❧❝✉❧♦s ❡r❛♠ ❢❡✐t♦s s❡♠ ❛ ✉t✐❧✐③❛çã♦ ❞♦ sí♠❜♦❧♦ ❞❡ ♣♦r❝❡♥t❛❣❡♠✱ ❡r❛♠ r❡❛❧✐③❛❞♦s ❞❡ ❢♦r♠❛ s✐♠♣❧❡s✱ ❝♦♠ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❢r❛çõ❡s ❝❡♥t❡s✐♠❛✐s✳ P♦r ❡①❡♠♣❧♦✱ ♥❛ ❝♦❜r❛♥ç❛ ❞❡ ✉♠ ✐♠♣♦st♦ ♥♦ ✈❛❧♦r ❞❡6/100❞❛ ❝♦♠❡r❝✐❛❧✐③❛çã♦✱ ❡❧❡s ❝♦❜r❛✈❛♠ s❡✐s
❝❡♥tés✐♠♦s ❞♦ ♣r❡ç♦ ❞♦ ♣r♦❞✉t♦✱ ✐st♦ é✱ ❞✐✈✐❞✐❛♠ ♦ ♣r♦❞✉t♦ ❡♠ ❝❡♠ ♣❛rt❡s ✐❣✉❛✐s ❡ ♣❡❣❛✈❛♠ s❡✐s ♣❛rt❡s✱ ❜❛s✐❝❛♠❡♥t❡ ♦ q✉❡ é ❢❡✐t♦ ❤♦❥❡✱ ♣♦ré♠ s❡♠ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❝❛❧❝✉❧❛❞♦r❛s✳
❆ ✐♥t❡♥s✐✜❝❛çã♦ ❞♦ ❝♦♠ér❝✐♦ ♣♦r ✈♦❧t❛ ❞♦ sé❝✉❧♦ ❳❱ ❝r✐♦✉ s✐t✉❛çõ❡s ❞❡ ❣r❛♥❞❡
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✹✳ P♦r❝❡♥t❛❣❡♠
♠♦✈✐♠❡♥t❛çã♦ ❝♦♠❡r❝✐❛❧✳ ❖ s✉r❣✐♠❡♥t♦ ❞♦s ❥✉r♦s✱ ❧✉❝r♦s ❡ ♣r❡❥✉í③♦s ♦❜r✐❣♦✉ ♦s ♠❛t❡♠át✐❝♦s ❛ ✜①❛r❡♠ ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ♣♦r❝❡♥t❛❣❡♥s✳ ❆ ❜❛s❡ ❡s❝♦❧❤✐❞❛ ❢♦✐ ♦100✳ ❖ ✐♥t❡r❡ss❛♥t❡ é q✉❡ ♠❡s♠♦ ❝♦♠ ❡ss❛ ❡✈♦❧✉çã♦✱ ♦ sí♠❜♦❧♦ q✉❡ ❝♦♥❤❡❝❡♠♦s
❤♦❥❡ ❛✐♥❞❛ ♥ã♦ ❡r❛ ✉t✐❧✐③❛❞♦ ♣❡❧♦s ❝♦♠❡r❝✐❛♥t❡s✳ ▼✉✐t♦s ❞♦❝✉♠❡♥t♦s ❡♥❝♦♥tr❛❞♦s ❡ r❡❣✐str❛❞♦s ❛♣r❡s❡♥t❛♠ ✉♠❛ ❢♦r♠❛ ❝✉r✐♦s❛ ❞❡ ❡①♣r❡ss❛r ♣♦r❝❡♥t❛❣❡♥s✳ ❖s r♦♠❛♥♦s ✉t✐❧✐③❛✈❛♠ ♦s ❛❧❣❛r✐s♠♦s ❞♦ s❡✉ s✐st❡♠❛ ❞❡ ♥✉♠❡r❛çã♦ s❡❣✉✐❞♦ ❞❡ s✐❣❧❛s ❝♦♠♦✱ ✏♣ ❝❡♥t♦✑ ❡ ✏♣ ❝✑✳ P♦r ❡①❡♠♣❧♦✱ ❛ ♣♦r❝❡♥t❛❣❡♠ ❞❡ 10% ❡r❛ ❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
✏❳ ♣ ❝❡♥t♦✑ ♦✉ ✏❳ ♣ ❝✑✱ ♦♥❞❡ X r❡♣r❡s❡♥t❛ 10❡♠ ❛❧❣❛r✐s♠♦s r♦♠❛♥♦s✳
❆ ❝r❡s❝❡♥t❡ ✉t✐❧✐③❛çã♦ ❞❛ ♣♦r❝❡♥t❛❣❡♠ ♥♦ ❝♦♠ér❝✐♦ ❡ ❛s s✉❛s ✐♥ú♠❡r❛s ❢♦r♠❛s ❞❡ ❡s❝r✐t❛ r❡♣r❡s❡♥t❛❝✐♦♥❛❧ ♦r✐❣✐♥❛r❛♠ ♦ sí♠❜♦❧♦ q✉❡ ❝♦♥❤❡❝❡♠♦s ❤♦❥❡ %✳ ✑
❈♦♠✉♠❡♥t❡ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ❡①♣r❡ssõ❡s ❞♦ t✐♣♦✿
• ♦ í♥❞✐❝❡ ❞❡ r❡❛❥✉st❡ ❞♦ s❛❧ár✐♦ ♠í♥✐♠♦ ❞❡ ❏❛♥❡✐r♦ ❞❡ 2013 ❢♦✐ ❞❡ 9%❀
• ❛ ✐♥✢❛çã♦ s✉♣❡r❛ ❛ ♠❡t❛ ❞❡ 4,5% ❡♠ 2012❀
• ✉♠❛ ❧♦❥❛ t❡♠ ✉♠ ❞❡s❝♦♥t♦ ❞❡ 30% s♦❜r❡ ♦ ✈❛❧♦r ❞❡ s❡✉s ♣r♦❞✉t♦s✳
❙❡rá q✉❡ ❡st❛♠♦s ❛♣t♦s ❛ ❝♦♠♣r❡❡♥❞❡r t❛✐s ✐♥❢♦r♠❛çõ❡s❄ P❛r❛ t❛♥t♦✱ é ♥❡❝❡ssár✐♦ ❢❛③❡r ✉♠ ❡st✉❞♦ t❡ór✐❝♦ s♦❜r❡ ♦ ❝♦♥❝❡✐t♦ ❜ás✐❝♦ s♦❜r❡ ♣♦r❝❡♥t❛❣❡♠✳ ➱ ♦ q✉❡ ❢❛r❡♠♦s ❛ s❡❣✉✐r✳ P♦r❝❡♥t❛❣❡♠ ✭♦✉ t❛①❛ ♣❡r❝❡♥t✉❛❧✮ é ✉♠ ♠♦❞♦ ❞❡ ❝♦♠♣❛r❛r ❣r❛♥❞❡③❛s ✉s❛♥❞♦ ♣r♦♣♦rçã♦ ❞✐r❡t❛✱ ✐st♦ é✱ ❡①♣r❡ss❛r ❛ r❛③ã♦ ❡♥tr❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ p❡ ♦ ♥ú♠❡r♦ 100✱
q✉❡ ❞❡♥♦♠✐♥❛♠♦s ♣♦r p%✭❧ê✲s❡ ✏p♣♦r ❝❡♥t♦✑✮✳
❯♠❛ ❞❛s ❛♣❧✐❝❛çõ❡s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❛ ✐❞❡✐❛ ❞❡ t❛①❛ ♣❡r❝❡♥t✉❛❧ sã♦ ❛s q✉❡ ❡♥✈♦❧✈❡♠ tr❛♥s❛çõ❡s ♠❡r❝❛♥t✐s ✭❝♦♠♣r❛s ❡ ✈❡♥❞❛s✮ q✉❡✱ ❜❛s✐❝❛♠❡♥t❡✱ ♣♦❞❡♠ ❣❡r❛r ❛❝rés❝✐♠♦s✱ ❞❡s❝♦♥t♦s✱ ❧✉❝r♦s ❡ ♣r❡❥✉í③♦s✱ ❝♦♠♦ ♥♦s ♠♦str❛♠ ♦s ♣ró①✐♠♦s ❡①❡♠♣❧♦s✳
❊①❡♠♣❧♦ ✼ ❋❡r♥❛♥❞♦ ✐♥✈❡st✐✉ R$10.000,00❡♠ ✉♠ ❢✉♥❞♦ ❞❡ ❛♣❧✐❝❛çã♦ ❡ ❤♦❥❡✱ ❛♣ós 5 ♠❡s❡s✱ ❡❧❡ t❡♠ R$11.500,00✳ ❱❛♠♦s ❞❡t❡r♠✐♥❛r ♦ s❡✉ r❡♥❞✐♠❡♥t♦ ♣❡r❝❡♥t✉❛❧
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✹✳ P♦r❝❡♥t❛❣❡♠
♥❡ss❡ ♣❡rí♦❞♦✳ ❈♦♠♦ s❡✉ ❣❛♥❤♦ ❢♦✐ ❞❡ R$1.500,00❡♠ r❡❧❛çã♦ ❛ ✉♠ ✐♥✈❡st✐♠❡♥t♦ ❞❡
R$10.000,00✱ t❡♠♦s q✉❡
1.500 10.000 =
p
100,
♦✉ s❡❥❛✱
p= 15%.
■st♦ q✉❡r ❞✐③❡r q✉❡ ❋❡r♥❛♥❞♦ ❣❛♥❤♦✉ 15% ❞♦ s❡✉ ✐♥✈❡st✐♠❡♥t♦ ✐♥✐❝✐❛❧ ❛♦ ✜♠ ❞♦
♣❡rí♦❞♦ ❞❡ 5 ♠❡s❡s✳
❊①❡♠♣❧♦ ✽ ❙❛❜❡♥❞♦ q✉❡ ♦ r❡❛❥✉st❡ ❞♦ s❛❧ár✐♦ ♠í♥✐♠♦ ❡♠ ❏❛♥❡✐r♦ ❞❡ 2013 ❢♦✐ ❞❡ 9% ❡♠ r❡❧❛çã♦ ❛♦s R$622,00 ❞❡ ✷✵✶✷✱ ✈❛♠♦s ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞♦ s❛❧ár✐♦ ♠í♥✐♠♦
q✉❡ ❛ ♣❛rt✐r ❞♦ ♠ês ♠❡♥❝✐♦♥❛❞♦ ❡♥tr♦✉ ❡♠ ✈✐❣♦r✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ✜③❡♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ t❡♠♦s
x
622 = 9 100,
♦♥❞❡ x é ♦ r❡❛❥✉st❡ ❞❡ 9% s♦❜r❡ R$622,00✳ ❋❛③❡♥❞♦ ♦s ❝á❧❝✉❧♦s✱ ♦❜t❡♠♦s q✉❡
x= 55,98
❡✱ ♣♦rt❛♥t♦✱ ♦ ♥♦✈♦ s❛❧ár✐♦ ♠í♥✐♠♦ é ✐❣✉❛❧ ❛ 622 + 55,98 = 677,98r❡❛✐s✳ ❚❡♥❞♦ ❡♠
✈✐st♦ ❛ ❛♣r♦①✐♠❛çã♦✱ ♦ ❣♦✈❡r♥♦ ❛❞♦t♦✉ q✉❡ ❡st❡ s❛❧ár✐♦ s❡r✐❛ ❞❡ R$678,00✳
❊①❡♠♣❧♦ ✾ ●❡r❛❧❞♦ ♣❛❣❛rá ❛ t❛①❛ ❞❡ ❝♦♥❞♦♠í♥✐♦ ❞♦ ♣ré❞✐♦ ♦♥❞❡ ♠♦r❛✱ q✉❡ ♥❡st❡ ♠ês é ❞❡ R$256,00 ❛♥t❡s ❞♦ ✈❡♥❝✐♠❡♥t♦ ♦❜t❡♥❞♦ ✉♠ ❞❡s❝♦♥t♦ ❞❡ 8% s♦❜r❡ ❡st❡
✈❛❧♦r✳ P♦❞❡♠♦s ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞♦ ❝♦♥❞♦♠í♥✐♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ❝❛❧❝✉❧❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ q✉❛♥t♦s r❡❛✐s ❝♦rr❡s♣♦♥❞❡♠ ❛ 8% ❞♦ ✈❛❧♦r ❞♦ ❝♦♥❞♦♠í♥✐♦✱ ♦✉ s❡❥❛✱
x
256 = 8 100,
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✹✳ P♦r❝❡♥t❛❣❡♠
♦♥❞❡ x é ♦ ✈❛❧♦r ❞♦ ❞❡s❝♦♥t♦ ❞❡ 8% s♦❜r❡ R$256,00✳ ❆ss✐♠✱
x= 20,48
❡ s✉❜tr❛✐♥❞♦ ♦ ✈❛❧♦r ♦❜t✐❞♦ ❞❛ t❛①❛ ❞♦ ❝♦♥❞♦♠í♥✐♦✱ t❡♠♦s q✉❡ ●❡r❛❧❞♦ ♣❛❣❛rá
256−20,48 = 235,52 r❡❛✐s✳
❈♦♠♦ ✈✐♠♦s ❡♠ ❛❧❣✉♠❛s s✐t✉❛çõ❡s✱ t❛♥t♦ ♦s ❛❝rés❝✐♠♦s ❝♦♠♦ ♦s ❞❡s❝♦♥t♦s ❝♦♥s✐❞❡r❛❞♦s ✐♥❝✐❞✐❛♠ s♦❜r❡ ♦ ✈❛❧♦r ✐♥✐❝✐❛❧✳ ❆❣♦r❛✱ ✐r❡♠♦s ❡st✉❞❛r ❛❧❣✉♠❛s s✐t✉❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ❛❝rés❝✐♠♦s ❡ ❞❡s❝♦♥t♦s s✉❝❡ss✐✈♦s✳ ❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s✳
❊①❡♠♣❧♦ ✶✵ ❙❡ ❛ t❛①❛ ❞❡ ✐♥✢❛çã♦ ❡♠ ❏❛♥❡✐r♦ ❢♦✐ ❞❡6% ❡ ❛ ❞❡ ❋❡✈❡r❡✐r♦ ❢♦✐ ❞❡5%✱
❡♥tã♦ ❛ t❛①❛ ❞❡ ✐♥✢❛çã♦ ❞❡ ❏❛♥❡✐r♦ ✴ ❋❡✈❡r❡✐r♦ ❢♦✐ ❞❡ 11,3%✳ ❉❡ ❢❛t♦✱ ❛ ✐♥✢❛çã♦
❛❝✉♠✉❧❛❞❛ ♥❡st❡ ♣❡rí♦❞♦ ❢♦✐ ❞❡
6% + 5% + (5% ❞❡ 6%),
♦✉ s❡❥❛✱ ❞❡ 11% + 0,3% = 11,3%✳
❆❝r❡s❝❡♥t❛r ♦✉ ❞❡s❝♦♥t❛r p% ❛ ✉♠❛ q✉❛♥t✐❛ q é ♠✉❧t✐♣❧✐❝❛r q ♣♦r ✉♠ ❢❛t♦r ❞❡
❝♦rr❡çã♦ f ❞❛❞♦ ♣♦r f = 1±p%✳
P❛r❛ ❝♦♠♣♦r ✈ár✐♦s ❛✉♠❡♥t♦s ❡✴♦✉ ❞❡s❝♦♥t♦s ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r ♦s ✈ár✐♦s ❢❛t♦r❡s ✐♥❞✐✈✐❞✉❛✐s ❡ ❛ss✐♠ ♦❜t❡r ♦ ❢❛t♦r ❛❝✉♠✉❧❛❞♦✱ q✉❡ ♥❛❞❛ ♠❛✐s é q✉❡ ♦ ❢❛t♦r ❞❡ ❛t✉❛❧✐③❛çã♦ ❡♥tr❡ ♦ ♣r✐♠❡✐r♦ ❡ ♦ ú❧t✐♠♦ ✈❛❧♦r ❝♦♥s✐❞❡r❛❞♦✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞♦s ✈❛❧♦r❡s ✐♥t❡r♠❡❞✐ár✐♦s✱ ✈❡❥❛ ❬✶✶❪✱
f❛❝✉♠✉❧❛❞♦ =f1·f2·f3·. . . .
◆♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ♣♦❞❡rí❛♠♦s t❡r ❝♦♥s✐❞❡r❛❞♦ f1 = 1 + 6% = 1 + 0,06 = 1,06❡
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✹✳ P♦r❝❡♥t❛❣❡♠
f2 = 1 + 5% = 1 + 0,05 = 1,05✳ ▲♦❣♦✱
f❛❝✉♠✉❧❛❞♦ =f1·f2 = 1,06·1,05 = 1 + 0,113 = 1 + 11,3%.
❈♦♠♣❛r❛♥❞♦✱ ♦❜t❡♠♦s 11,3% ❝♦♠♦ ❛ ✐♥✢❛çã♦ ❛❝✉♠✉❧❛❞❛ ♥❡st❡ ♣❡rí♦❞♦✳
❊①❡♠♣❧♦ ✶✶ ❊♠ ✉♠❛ ❧✐q✉✐❞❛çã♦✱ ♦s ♣r❡ç♦s ❞♦s ❛rt✐❣♦s ❞❡ ✉♠❛ ❧♦❥❛ sã♦ r❡❞✉③✐❞♦s ❡♠ 20% ❞♦ s❡✉ ✈❛❧♦r✳ ❚❡r♠✐♥❛❞❛ ❡st❡ ❧✐q✉✐❞❛çã♦✱ ❡ ♣r❡t❡♥❞❡♥❞♦ ✈♦❧t❛r ❛♦s ♣r❡ç♦s
♦r✐❣✐♥❛✐s✱ ❡♠ q✉❡ ♣♦r❝❡♥t❛❣❡♠ ❞❡✈❡♠ s❡r ❛✉♠❡♥t❛❞♦s ♦s ♣r❡ç♦s ❞❛ ❧✐q✉✐❞❛çã♦❄ ❈♦♠♦ ♥♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ✉s❛r❡♠♦s ♦ ❢❛t♦r ❞❡ ❝♦rr❡çã♦✳ ❚❡♠♦s q✉❡
f1 = 1−20% = 1−0,2 = 0,8 ❡ f2 = 1 +p%,
♦♥❞❡ p% é ♦ r❡❛❥✉st❡ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ✈♦❧t❛ ❞♦s ♣r❡ç♦s ♦r✐❣✐♥❛✐s✳ ▲♦❣♦✱ ♦ ❢❛t♦r
❛❝✉♠✉❧❛❞♦ é ✐❣✉❛❧ ❛ 1 ❡✱ ❛ss✐♠✱
1 = 0,8·(1 +p%) ⇒ p% = 25%.
❊①❡♠♣❧♦ ✶✷ ❆ t❛❜❡❧❛ ❛ s❡❣✉✐r ♠♦str❛ ❛ ✈❛r✐❛çã♦ ❞♦ ♣r❡ç♦ ❞♦ ❞ó❧❛r ❡♠ ✉♠❛ s❡♠❛♥❛ q✉❛❧q✉❡r ❡♠ t❡r♠♦s ♣❡r❝❡♥t✉❛✐s✳ ◆♦ ✈❛❧♦r ❛❝✉♠✉❧❛❞♦ ❞❡ss❡s 5❞✐❛s✱ ♦ q✉❡ ❛❝♦♥t❡❝❡✉
❝♦♠ ♦ ♣r❡ç♦ ❞♦ ❞ó❧❛r❄ ❙❡rá q✉❡ s✉❜✐✉ ♦✉ ❜❛✐①♦✉❄ ❊♠ q✉❛♥t♦s ♣♦r ❝❡♥t♦❄
❉■❆
❱❆❘■❆➬➹❖
❙❡❣✉♥❞❛✲❢❡✐r❛
✲✷✱✸✺ ✪
❚❡rç❛✲❢❡✐r❛
✶✱✸✼ ✪
◗✉❛rt❛✲❢❡✐r❛
✶✱✵✺ ✪
◗✉✐♥t❛✲❢❡✐r❛
✲✵✱✶✸✪
❙❡①t❛✲❢❡✐r❛
✵✱✷✶✪
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✳✹✳ P♦r❝❡♥t❛❣❡♠
❆♥❛❧♦❣❛♠❡♥t❡ ❛♦ q✉❡ ✜③❡♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ t❡♠♦s q✉❡
f1 = 1−0,0235 = 0,9765, f2 = 1 + 0,0137 = 1,0137,
f3 = 1 + 0,0105 = 1,0105, f4 = 1−0,0013 = 0,9987,
f5 = 1 + 0,0021 = 1,0021.
P♦rt❛♥t♦✱ ❢❛③❡♥❞♦ ♦s ❝á❧❝✉❧♦s✱ ♦❜t❡♠♦s q✉❡
f❛❝✉♠✉❧❛❞♦ = f1·f2·f3·f4·f5
= 0,9765·1,0137·1,0105·0,9987·1,0021
≃ 1,00107.
❆ss✐♠✱ ❝♦♠♦ ♦ ❢❛t♦r ❛❝✉♠✉❧❛❞♦ é s✉♣❡r✐♦r ❛1✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♥❡st❡s5❞✐❛s✱ ♦ ♣r❡ç♦
❞♦ ❞ó❧❛r s✉❜✐✉ ❡♠ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 0,107%✳
❈❛♣ít✉❧♦ ✷
▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛
◆❡st❡ ❝❛♣ít✉❧♦ ✐❞❡♥t✐✜❝❛r❡♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ✈❛❧♦r ❞❡ ✉♠ ❝❛♣✐t❛❧ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦✱ ❝♦♥❝❡✐t✉❛♥❞♦ ✐♥str✉♠❡♥t♦s q✉❡ sã♦ r❡❣✉❧❛♠❡♥t❛❞♦s ♣❡❧♦ ♠❡r❝❛❞♦ ✜♥❛♥❝❡✐r♦✱ ❡①❡♠♣❧✐✜❝❛♥❞♦✲♦s ❡♠ s✐t✉❛çõ❡s ❞♦ ♥♦ss♦ ❝♦t✐❞✐❛♥♦✳
✷✳✶ ❖♣❡r❛çõ❡s ❈♦♠❡r❝✐❛✐s
❖♣❡r❛çõ❡s ❝♦♠❡r❝✐❛✐s sã♦ tr❛♥s❛çõ❡s ❞❡ ❝♦♠♣r❛ ♦✉ ❞❡ ✈❡♥❞❛✱ ❢❡✐t❛s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♦❜t❡r ❧✉❝r♦✳
❆ ✈❡♥❞❛ ❝♦♥s✐st❡ ♥♦ tr❡s♣❛ss❡✶ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ✉♠ ❜❡♠ q✉❛♥❞♦ ♣❛❣♦ ✉♠
✈❛❧♦r ❡st✐♣✉❧❛❞♦✳ ❊ss❡ ❞❡t❡r♠✐♥❛❞♦ ✈❛❧♦r é ❝❤❛♠❛❞♦ ❞❡ ♣r❡ç♦ ❞❡ ❝✉st♦ ♦✉ ♣r❡ç♦ ❞❡ ✈❡♥❞❛✱ q✉❡ ♥♦ ❝♦t✐❞✐❛♥♦ é✱ ❣❡r❛❧♠❡♥t❡✱ ❡♥t❡♥❞✐❞♦ ❝♦♠♦ t❡♥❞♦ ♦ ♠❡s♠♦ s✐❣♥✐✜❝❛❞♦✳ ❊♥tr❡t❛♥t♦✱ ♥❡st❡ tr❛❜❛❧❤♦✱ ❞✐❢❡r❡♥❝✐❛♠♦s ❡ss❡s ❞♦✐s ❝♦♥❝❡✐t♦s ❝♦♠♦ s❡❣✉❡✿
• ♣r❡ç♦ ❞❡ ❝✉st♦✿ é ♦ ✈❛❧♦r ♣❛❣♦ ♣❛r❛ ❛q✉✐s✐çã♦ ❞♦ ♣r♦❞✉t♦ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡
♦❜t❡r ❧✉❝r♦✳
• ♣r❡ç♦ ❞❡ ✈❡♥❞❛✿ é ♦ ✈❛❧♦r ❝♦❜r❛❞♦ ❛♦ ❝♦♥s✉♠✐❞♦r✳
✶➱ ✉♠ ❝♦♥tr❛t♦ ❞❡ ❝♦♠♣r❛ ♦✉ ✈❡♥❞❛ ♣❡❧♦ q✉❛❧ ♦❝♦rr❡ ❛ tr❛♥s❢❡rê♥❝✐❛ ❞❡ t✐t✉❧❛r✐❞❛❞❡✳
