❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❙♦❜r❡ ❏✉r♦s ❡ ❆♣❧✐❝❛çã♦ ❞❡ ❈♦♥❝❡✐t♦s ❈❧áss✐❝♦s
❡♠ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛
❯❧②ss❡s ❖r❧❛♥❞♦ ❏ú♥✐♦r
❇r❛sí❧✐❛
❯❧②ss❡s ❖r❧❛♥❞♦ ❏ú♥✐♦r
❙♦❜r❡ ❏✉r♦s ❡ ❆♣❧✐❝❛çã♦ ❞❡ ❈♦♥❝❡✐t♦s
❈❧áss✐❝♦s ❡♠ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❑❡❧❧❝✐♦ ❖❧✐✈❡✐r❛ ❆r❛✉❥♦
Ficha catalográfica elaborada automaticamente, com os dados fornecidos pelo(a) autor(a)
O71s
Orlando Júnior, Ulysses
Sobre Juros e Aplicação de Conceitos Clássicos em Matemática Financeira / Ulysses Orlando Júnior; orientador Kellcio Oliveira Araujo. -- Brasília, 2015. 76 p.
Dissertação (Mestrado - Mestrado Profissional em Matemática) -- Universidade de Brasília, 2015.
❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳ ➱ ♣r♦✐❜✐❞❛ ❛ r❡♣r♦❞✉çã♦ t♦t❛❧ ♦✉ ♣❛r❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ s❡♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❞♦ ❛✉t♦r ❡ ❞♦ ♦r✐❡♥t❛❞♦r✳
❆ ♠❛t❡♠át✐❝❛ é ❛ ú♥✐❝❛ ❧✐♥❣✉❛❣❡♠ q✉❡ t❡♠♦s ❡♠ ❝♦♠✉♠ ❝♦♠ ❛ ♥❛t✉r❡③❛✳
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ q✉❡ s❡♠♣r❡ ♠❡ ❛❜❡♥ç♦♦✉ ❡ ♠❡ ❛❣r❛❝✐♦✉ ❝♦♠ ♠❛✐s ❡ss❛ ♦♣♦r✲ t✉♥✐❞❛❞❡✱ q✉❡ ♠❡ ❞❡✉ ❢♦rç❛ ❡ ✐❧✉♠✐♥❛çã♦ ❡♠ t♦❞♦ ♦ tr❛❥❡t♦ ♣❡r❝♦rr✐❞♦✱ ♣♦ss✐❜✐❧✐t❛♥❞♦ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ ❝✉rs♦✱ tã♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛✳
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♠✐♥❤❛ ❡s♣♦s❛ ❙✐❧✈❛♥❛ ❡ ♠❡✉s ✜❧❤♦s ❘❡♥❛t❛✱ ❇r✉♥❛✱ ▲✉❝❛s✱ ●❛✲ ❜r✐❡❧ ❡ ❘❛❢❛❡❧ q✉❡ ♠✉✐t♦ ♠❡ ♠♦t✐✈❛♠ ♥❛ ❜✉s❝❛ ❡♠ t♦r♥❛r✲♠❡ ✉♠ s❡r ❤✉♠❛♥♦ ♠❡❧❤♦r✱ ❡ ❡♠ s❡r ✉♠ ❡s♣♦s♦ ❡ ♣❛✐ à ❛❧t✉r❛ ❞♦ ❛♠♦r q✉❡ s✐♥t♦ ❡ ❛♦s ♠❡✉s ♣❛✐s✱ ❢♦♥t❡ ♠❛✐♦r ❞❡ ✐♥s♣✐r❛çã♦✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ❉r✳ ❑❡❧❧❝✐♦ ❖❧✐✈❡✐r❛ ❆r❛✉❥♦✱ ♣❡❧❛s ❝♦rr❡çõ❡s✱ ❝♦♥s❡❧❤♦s✱ ♦r✐❡♥t❛çõ❡s ❡ ❝♦♥tr✐❜✉✐çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❡st❡ tr❛❜❛❧❤♦✳
❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❡♠ ❡s♣❡❝✐❛❧ àq✉❡❧❡s q✉❡ ♠✐♥✐s✲ tr❛r❛♠ ❝✉rs♦s ♣❛r❛ ❛ t✉r♠❛ ❞❡ ✷✵✶✸✱ ❜❡♠ ❝♦♠♦ ❛♦s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❡st✐✈❡r❛♠ ❡♥✈♦❧✈✐❞♦s ❝♦♠ ♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ P❘❖❋▼❆❚✱ ♣❡❧♦s ♠♦♠❡♥✲ t♦s ❞❡ ♣❛rt✐❧❤❛ ❡ ❝♦♠♣❛♥❤❡✐r✐s♠♦ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ q✉❡ ❡str❡✐táss❡♠♦s ❧❛ç♦s ❞❡ ❛♠✐③❛❞❡✳ ❉❡ ♠♦❞♦ ❡s♣❡❝✐❛❧ ❛♦s ❛♠✐❣♦s ❊♠♠❛♥✉❡❧✱ ❊♠❡rs♦♥✱ ●✉st❛✈♦ ❡ ❘✐❝❛r❞♦ q✉❡ t❛♥t❛s t❛r❞❡s ❡ ♥♦✐t❡s ♣❛ss❛♠♦s ❛♦ ❧♦♥❣♦ ❞❡ ✷✵✶✸ ❛ ✷✵✶✺ ♥♦s ❞❡❞✐❝❛♥❞♦ ❛ ❝✉♠♣r✐r ❛s ❣r❛t✐✜❝❛♥t❡s ❡t❛♣❛s ❞❡ss❡ ❝✉rs♦ ❞❡ ♠❡str❛❞♦✳
➚ ❈❆P❊❙✱ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ❛❜♦r❞❛♠♦s ♦ ❝á❧❝✉❧♦ ❞❡ ❥✉r♦s ❛♣❧✐❝❛çõ❡s ❞❡ ❝♦♥❝❡✐t♦s ❝❧áss✐❝♦s ❞❛ ♠❛t❡♠át✐❝❛ ❜ás✐❝❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❢r❛çõ❡s✱ ♥ú♠❡r♦s ❞❡❝✐♠❛✐s✱ r❡❣r❛ ❞❡ três✱ ❢✉♥çõ❡s ❛✜♠ ❡ ❡①♣♦♥❡♥❝✐❛❧✱ ❜❡♠ ❝♦♠♦ ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ❡ ❣❡♦♠étr✐❝❛s✱ q✉❡ s❡ r❡✈❡❧❛♠ ❝♦♠♦ ❢❡rr❛♠❡♥t❛s ❡✜❝❛③❡s ♥❛ ❛♣❧✐❝❛çã♦ ❞♦ ❝á❧❝✉❧♦ ❞❡ ❥✉r♦s ❡ ♣♦r❝❡♥t❛❣❡♥s ❡ ♥❛ r❡s♦✲ ❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞♦ ❛♠❜✐❡♥t❡ ❡s❝♦❧❛r✱ t❛♥t♦ ♣❛r❛ ♦ ❛❧✉♥♦ ❝♦♠♦ ♣❛r❛ ♦ ♣r♦❢❡ss♦r✳ ❆♣r❡s❡♥t❛♠♦s t❛✐s ❝♦♥❝❡✐t♦s ❝♦♠♦ ❜❛s❡ ❞♦s ❝♦♥❝❡✐t♦s ❞❛ ♠❛t❡♠át✐❝❛ ✜♥❛♥❝❡✐r❛ ♣❛r❛ ❢❛❝✐❧✐t❛r ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ✐♠♣❧❡♠❡♥t❛r ❛ ❡❞✉❝❛çã♦ ✜♥❛♥❝❡✐r❛ q✉❡ é ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❡❞✉❝❛t✐✈♦ q✉❡ ❛♣❧✐❝❛ ♠ét♦❞♦s ♣ró♣r✐♦s✱ ❡♠ q✉❡ ♦ ♣r♦❢❡ss♦r ❛✉①✐❧✐❛ ♦ ❛❧✉♥♦ ❛ ❝♦♥str✉✐r s✉❛ ❛✉t♦♥♦♠✐❛ ♣❛r❛ ❛♥❛❧✐s❛r ❡ ❛r❣✉♠❡♥t❛r s♦❜r❡ ✜♥❛♥ç❛s✱ ❞❡s❡♥✈♦❧✈❡r ❛t✐✈✐❞❛❞❡s ♣❛r❛ ❛✉①✐❧✐❛r ♦s ❝♦♥s✉♠✐❞♦r❡s ❛ ♦rç❛r ❡ ❣❡r✐r ❛ s✉❛ r❡♥❞❛✱ ❛ ♣♦✉♣❛r ❡ ❛ ✐♥✈❡st✐r✳ ❙ã♦ ❡❧❡♠❡♥✲ t♦s ❡ ❝♦♥❝❡♣çõ❡s s✐❣♥✐✜❝❛t✐✈❛s ♣❛r❛ q✉❡ ♦ ❝✐❞❛❞ã♦ ❡①❡rç❛ ❛t✐✈✐❞❛❞❡✱ tr❛❜❛❧❤♦✱ ♣r♦✜ssã♦ ❡ ❧❛③❡r✱ ❛♥❛❧✐s❛♥❞♦ ❡ ❛r❣✉♠❡♥t❛♥❞♦ ❞✐❛♥t❡ ❞♦s ❛♣❡❧♦s ❞♦ ❝♦♥s✉♠✐s♠♦✳ ❆♦ ❛ss♦❝✐❛r ♥♦çõ❡s ❞❡ ❊❝♦♥♦♠✐❛ ❝♦♠ ❝♦♥t❡ú❞♦s ❞❡ ▼❛t❡♠át✐❝❛✱ ❢♦❝❛♥❞♦ ❛ ▼❛t❡♠át✐❝❛ ❇ás✐❝❛ ❛♣❧✐❝❛❞❛ ♥❛ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ♥♦ ❡♥s✐♥♦ ❛ ✐♥t❡♥çã♦ é ♠♦str❛r ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♠❡❧❤♦r❛r ❛ ♣r♦❜❧❡♠át✐❝❛ q✉❡ r❡s✐❞❡ ♥♦ ♣❛♥♦r❛♠❛ ✜♥❛♥❝❡✐r♦ ❞♦s ❡st✉❞❛♥t❡s✳
P❛❧❛✈r❛s✲❝❤❛✈❡
▼❛t❡♠át✐❝❛ ❇ás✐❝❛❀ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛❀ ❊❞✉❝❛çã♦ ❋✐♥❛♥❝❡✐r❛❀ P♦✉♣❛r❀ ■♥✈❡s✲ t✐r❀ ❈✐❞❛❞❛♥✐❛✳
❆❜str❛❝t
■♥ t❤✐s ♣❛♣❡r ✇❡ r❡♣♦rt t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ❛♣♣❧✐❝❛t✐♦♥s ✐♥t❡r❡st ♦❢ ❝❧❛ss✐❝ ❝♦♥❝❡♣ts ♦❢ ❜❛s✐❝ ♠❛t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢r❛❝t✐♦♥s✱ ❞❡❝✐♠❛❧ ♥✉♠❜❡rs✱ t❤r❡❡ r✉❧❡ ✐♥ ♦r❞❡r ❢✉♥❝t✐♦♥s✱ ❡①♣♦♥❡♥t✐❛❧✱ ❛♥❞ ❣❡♦♠❡tr✐❝ ❛♥❞ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥✱ t❤❛t r❡✈❡❛❧ t❤❡♠s❡❧✈❡s ❛s ❡❢✲ ❢❡❝t✐✈❡ t♦♦❧s ✐♥ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ✐♥t❡r❡st ❝❛❧❝✉❧❛t✐♦♥ ❛♥❞ ♣❡r❝❡♥t❛❣❡s ❛♥❞ r❡s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ♦❢ t❤❡ s❝❤♦♦❧ ❡♥✈✐r♦♥♠❡♥t✱ ❜♦t❤ ❢♦r t❤❡ st✉❞❡♥t ❛♥❞ t❤❡ t❡❛❝❤❡r✳ ■♥tr♦❞✉❝✐♥❣ s✉❝❤ ❝♦♥❝❡♣ts ❛s t❤❡ ❜❛s✐s ♦❢ t❤❡ ❝♦♥❝❡♣ts ♦❢ ✜♥❛♥❝✐❛❧ ♠❛t❤❡♠❛t✐❝s t♦ ❢❛❝✐❧✐t❛t❡ ✉♥❞❡rs✲ t❛♥❞✐♥❣ ❛♥❞ ✐♠♣❧❡♠❡♥t✐♥❣ ✜♥❛♥❝✐❛❧ ❡❞✉❝❛t✐♦♥ ✇❤✐❝❤ ✐s ❛♥ ❡❞✉❝❛t✐♦♥❛❧ ♣r♦❝❡❞✉r❡ t❤❛t ❛♣♣❧✐❡s ♦✇♥ ♠❡t❤♦❞s✱ ✐♥ ✇❤✐❝❤ t❤❡ t❡❛❝❤❡r ❤❡❧♣s t❤❡ st✉❞❡♥t t♦ ❜✉✐❧❞ t❤❡✐r ❛✉t♦♥♦♠② t♦ ❛♥❛❧②③❡ ❛♥❞ ❛r❣✉❡ ❛❜♦✉t ✜♥❛♥❝❡s✱ ❞❡✈❡❧♦♣ ❛❝t✐✈✐t✐❡s t♦ ❤❡❧♣ ❝♦♥s✉♠❡rs t♦ ❜✉❞❣❡t ❛♥❞ ♠❛♥❛❣❡ ②♦✉r ✐♥❝♦♠❡✱ s❛✈✐♥❣ ❛♥❞ ✐♥✈❡st✐♥❣✳ ❚❤❡② ❛r❡ s✐❣♥✐✜❝❛♥t ❡❧❡♠❡♥ts ❛♥❞ ❝♦♥❝❡♣ts s♦ t❤❛t ❝✐t✐③❡♥s ❡①❡r❝✐s❡ ❛❝t✐✈✐t②✱ ❥♦❜ ❛♥❞ ❧❡✐s✉r❡✱ ❛♥❛❧②③✐♥❣ ❛♥❞ ❛r❣✉✐♥❣ ❛♣♣❡❛❧s ❜❡❢♦r❡ ❝♦♥s✉♠❡r✐s♠✳ ❇② ❛ss♦❝✐❛t✐♥❣ ♥♦t✐♦♥s ♦❢ ❡❝♦♥♦♠✐❝s ✇✐t❤ ♠❛t❤❡♠❛t✐❝s ❝♦♥t❡♥t✱ ❢♦❝✉s✐♥❣ ♦♥ ❇❛s✐❝ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s ✐♥ t❡❛❝❤✐♥❣ t❤❡ ✐♥t❡♥t✐♦♥ ✐s t♦ s❤♦✇ ♣♦ss✐❜✐❧✐t✐❡s t♦ ✐♠♣r♦✈❡ t❤❡ ♣r♦❜❧❡♠ ❧✐❡s ✐♥ t❤❡ ✜♥❛♥❝✐❛❧ ♦✉t❧♦♦❦ ♦❢ st✉❞❡♥ts✳ ❑❡②✇♦r❞s
❇❛s✐❝ ▼❛t❤❡♠❛t✐❝s❀ ❋✐♥❛♥❝✐❛❧ ♠❛t❤❀ ❋✐♥❛♥❝✐❛❧ ❡❞✉❝❛t✐♦♥❀ ❙❛✈❡❀ ■♥✈❡st✐♥❣❀ ❈✐t✐✲ ③❡♥s❤✐♣✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶ ●rá✜❝♦ ❞❛ ❋✉♥çã♦ ❊①♣♦♥❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ Pr♦❣r❡ssã♦ ❆r✐t♠ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ Pr♦❣r❡ssã♦ ●❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✶ ❆♣r❡s❡♥t❛çã♦ ❞❡ ❚❛①❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷ ❱❛♥t❛❣❡♥s ❡ ❉❡s✈❛♥t❛❣❡♥s ❞♦ ❈❛rtã♦ ❞❡ ❈ré❞✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✺
✷ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ✶✽
✷✳✶ ❆s ❢r❛çõ❡s ❡ ♦s ♥ú♠❡r♦s ❞❡❝✐♠❛✐s ♥♦ ❝á❧❝✉❧♦ ❞❛s ♣♦r❝❡♥t❛❣❡♥s ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✳✶ ❚r❛♥s❢♦r♠❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ❆ r❡❣r❛ ❞❡ três ♥♦ ❝á❧❝✉❧♦ ❞❛s ♣♦r❝❡♥t❛❣❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✸ ❈♦♥❝❡✐t♦s ❈❧áss✐❝♦s ❞❛ ▼❛t❡♠át✐❝❛ ❛♣❧✐❝❛❞♦s à ▼❛t❡♠át✐❝❛ ❋✐♥❛♥✲
❝❡✐r❛ ✷✼
✸✳✶ ❆ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✶✳✶ P♦r❝❡♥t❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✷ ❋✉♥çã♦ ❆✜♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✷✳✶ ❋✉♥çã♦ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✸ ❋✉♥çã♦ ❊①♣♦♥❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✹ Pr♦❣r❡ssõ❡s ❆r✐t♠ét✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✹✳✶ ❚❡r♠♦ ●❡r❛❧ ❞❡ ✉♠❛ Pr♦❣r❡ssã♦ ❆r✐t♠ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✹✳✷ ❙♦♠❛ ❞♦s ♥ ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ✉♠❛ Pr♦❣r❡ssã♦ ❆r✐t♠ét✐❝❛ ✳ ✳ ✸✽ ✸✳✺ Pr♦❣r❡ssõ❡s ●❡♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✺✳✶ ❚❡r♠♦ ●❡r❛❧ ❞❡ ✉♠❛ Pr♦❣r❡ssã♦ ●❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✺✳✷ ❙♦♠❛ ❞♦s ♥ ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ✉♠❛ Pr♦❣r❡ssã♦ ●❡♦♠étr✐❝❛ ✳ ✳ ✹✸ ✸✳✻ ❏✉r♦s ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✻✳✶ ❋ór♠✉❧❛s ❞❡ ❏✉r♦s ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✻✳✷ ❋ór♠✉❧❛s ❞♦ ▼♦♥t❛♥t❡ ♦✉ ❱❛❧♦r ❋✉t✉r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✻✳✸ ❋ór♠✉❧❛ ❞♦ ❈❛♣✐t❛❧ ♦✉ ❱❛❧♦r Pr❡s❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✼ ❏✉r♦s ❈♦♠♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✽ ❘❡s♦❧✉çõ❡s ❞❡ ♣r♦❜❧❡♠❛s ❛♣❧✐❝❛♥❞♦ ❝♦♥❝❡✐t♦s ❝❧áss✐❝♦s ❞❛ ▼❛t❡♠át✐❝❛
❇ás✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
✹ ❊❞✉❝❛çã♦ ❋✐♥❛♥❝❡✐r❛ ✈♦❧t❛❞❛ ♣❛r❛ ❥♦✈❡♥s ✺✼
✹✳✹✳✶ ❈✐❧❛❞❛s ❝♦♠ ❈❛rtã♦ ❞❡ ❈ré❞✐t♦ ♦✉ ❈❤❡q✉❡ ❊s♣❡❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
✺ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✼✸
❘❡❢❡rê♥❝✐❛s ✼✺
✶ ■♥tr♦❞✉çã♦
❆❧❣✉♥s ❡s♣❡❝✐❛❧✐st❛s ❡♠ ❊❞✉❝❛çã♦ ❋✐♥❛♥❝❡✐r❛ ❝♦♥❢♦r♠❡ ❬✼❪✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♥t❡♥❞❡♠ q✉❡ ❛ ♦♠✐ssã♦ ❞❛s ❡s❝♦❧❛s ❡♠ r❡❧❛çã♦ ❛ ♥♦çõ❡s ❞♦ ❝♦♠ér❝✐♦✱ ❡❝♦♥♦♠✐❛✱ ❞❡ ✐♠♣♦st♦s ❡ ❞❡ ✜♥❛♥ç❛s t❡♠ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ♣❡r✈❡rs❛✿ ❛ ♠❛✐♦r✐❛ ❞❛s ♣❡ss♦❛s✱ q✉❛♥❞♦ ❛❞✉❧t❛✱ ❝♦♥✲ t✐♥✉❛ ✐❣♥♦r❛♥❞♦ ❡ss❡s ❛ss✉♥t♦s ❡ s❡❣✉❡ s❡♠ ✐♥str✉çã♦ ✜♥❛♥❝❡✐r❛ ❡ s❡♠ ❤❛❜✐❧✐❞❛❞❡ ♣❛r❛ ♠❛♥❡❥❛r ❞✐♥❤❡✐r♦✳ ❆s ❝♦♥s❡q✉ê♥❝✐❛s s❡ t♦r♥❛♠ ♠❛✐s ❣r❛✈❡s✱ ♣♦✐s ♥✐♥❣✉é♠✱ q✉❛❧q✉❡r q✉❡ s❡❥❛ ❛ s✉❛ ♣r♦✜ssã♦✱ ❡stá ❧✐✈r❡ ❞♦s ♣r♦❜❧❡♠❛s ❧✐❣❛❞♦s ❛♦ ♠✉♥❞♦ ❞❛s ✜♥❛♥ç❛s ♣❡ss♦✲ ❛✐s✳ ➱ ✐♠♣♦rt❛♥t❡ t♦♠❛r ❝♦♥s❝✐ê♥❝✐❛ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛❧❢❛❜❡t✐③❛çã♦ ✜♥❛♥❝❡✐r❛✱ ♦ q✉❡ ♣♦❞❡ ♦❝♦rr❡r ♣♦r ✐♥✐❝✐❛t✐✈❛ ♣ró♣r✐❛✱ ♣♦r ♦r✐❡♥t❛çã♦ ❞♦s ♣❛✐s ♦✉ ♣♦r ❝♦♥s❡❧❤♦s ❞❡ ❛♠✐❣♦s✳
■♥❢❡❧✐③♠❡♥t❡✱ ♣❛r❛ ♠✉✐t❛s ♣❡ss♦❛s ♦ ❛❧❡rt❛ ❝❤❡❣❛ ❡♠ ✈✐rt✉❞❡ ❞❡ ❛❧❣✉♠ ❞❡s❛str❡ ✜♥❛♥✲ ❝❡✐r♦ q✉❡ ♥❛ ♠❛✐♦r✐❛ ❞♦s ❝❛s♦s✱ ❞❡❝♦rr❡ ❞❛ ❢❛❧t❛ ❞❛ ❡❞✉❝❛çã♦ ✜♥❛♥❝❡✐r❛✱ ❝♦♠♦ t❛♠❜é♠ ❞❛ ❢❛❧t❛ ❞❡ ❝♦♥tr♦❧❡ ❡ ♣❧❛♥❡❥❛♠❡♥t♦ ✜♥❛♥❝❡✐r♦✱ ❣❡r❛♥❞♦ ♣r♦❜❧❡♠❛s ♥♦ ♦rç❛♠❡♥t♦ ♣❡s✲ s♦❛❧✳
❊①✐st❡♠ t❛♠❜é♠ ♦✉tr♦s ❢❛t♦r❡s q✉❡ ❛❢❡t❛♠ ♦ ♦rç❛♠❡♥t♦ ❞♦♠ést✐❝♦✱ sã♦ ❡①trí♥s❡❝♦s ❡ ❢♦❣❡♠ ❛♦ ❝♦♥tr♦❧❡ ❡ ♣❧❛♥❡❥❛♠❡♥t♦ ❞❛s ♣❡ss♦❛s✳ ➱ ♦ ❝❛s♦ ❞❛s ❝r✐s❡s ❡❝♦♥ô♠✐❝❛s ❡ s✉❛s ❝♦♥s❡q✉ê♥❝✐❛s s♦❜r❡ ❡♠♣r❡❣♦ ❡ r❡♥❞❛✳ ❊ss❡s ❡✈❡♥t♦s ♣♦❞❡♠ ❡ ❞❡✈❡♠✱ ❡♠ ♠✉✐t♦s ❝❛s♦s✱ s❡r ♠✐♥✐♠✐③❛❞♦s✱ ❞❡s❞❡ q✉❡ s❡ ❛❞♦t❡♠ ❛❧❣✉♠❛s ♣r♦✈✐❞ê♥❝✐❛s ♣r❡✈❡♥t✐✈❛s✱ ❝❛s♦ ❝♦♥trár✐♦✱ ♣♦❞❡ ❤❛✈❡r t✉r❜✉❧ê♥❝✐❛ ♣❛ss❛❣❡✐r❛ ♥♦ ❡q✉✐❧í❜r✐♦ ✜♥❛♥❝❡✐r♦✱ ♠❛s q✉❡✱ ❞❡♥tr♦ ❞❡ ✉♠ ♣r♦❝❡ss♦ ❞❡ ♣❧❛♥❡❥❛♠❡♥t♦ ✜♥❛♥❝❡✐r♦✱ ❝♦♥s❡❣✉❡♠ s❡r r❛③♦❛✈❡❧♠❡♥t❡ ❝♦♥t♦r♥❛❞❛s✳
▼❡s♠♦ ❝♦♠ ✉♠ ♥í✈❡❧ ❞❡ ✐♥✢❛çã♦ ♠❛✐s ❜❛✐①♦ ♥♦s ❞✐❛s ❞❡ ❤♦❥❡✱ ❛ ❛❞♠✐♥✐str❛çã♦ ❞♦ ♦rç❛♠❡♥t♦ ❞♦♠ést✐❝♦ ❡①✐❣❡ ✈❡r❞❛❞❡✐r♦s ♠❛❧❛❜❛r✐s♠♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❛r❛ q✉❡♠ ❥á ♦♣❡r❛ ♥♦s ❧✐♠✐t❡s ❞❡ s❡✉s r❡♥❞✐♠❡♥t♦s✱ ❝❛s♦ ❞❛ ♠❛✐♦r✐❛ ❞❛s ♣❡ss♦❛s✳
P❛r❛ t❛♥t♦✱ ❡s❝♦❧❤❡✉✲s❡ ❛❜♦r❞❛r ✉♠ t❡♠❛ ❞❡ ✐♠♣♦rtâ♥❝✐❛ r❡❝♦♥❤❡❝✐❞❛ ♣❛r❛ t♦❞❛s ❛s ❢❛♠í❧✐❛s✱ ✏❙♦❜r❡ ❏✉r♦s ❡ ❆♣❧✐❝❛çã♦ ❞❡ ❈♦♥❝❡✐t♦s ❈❧áss✐❝♦s ❡♠ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛✑✱ q✉❡ t❡♥❞❡ ❛ ❛❥✉❞❛r ❛s ♣❡ss♦❛s ❛ ♦❜t❡r❡♠ ❝♦♥tr♦❧❡ ❞♦ ♦rç❛♠❡♥t♦ ♣❡ss♦❛❧ ❡ ❢❛♠✐❧✐❛r✱ ❡✈✐t❛r❡♠ ❛s ❢✉t✉r❛s ❞í✈✐❞❛s ♣❡ss♦❛✐s✱ ✐❞❡♥t✐✜❝❛r ❛s ♠❡❧❤♦r❡s ♦♣çõ❡s ❞❡ ✐♥✈❡st✐♠❡♥t♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❝❛❞❛ ♦❜❥❡t✐✈♦✱ ❡ ❡♥t❡♥❞❡r❡♠ ❛❧❣✉♠❛s ❧✐♥❤❛s ❞❡ ❝ré❞✐t♦s ♦❢❡r❡❝✐❞❛s ♣❡❧♦ ♠❡r❝❛❞♦✳
❈♦♠♦ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡st❡ ❡st✉❞♦✱ t❡♠✲s❡ q✉❡✱ ❡♠❜♦r❛ s❡ ♦❜s❡r✈❡♠ ❞✐✈❡rs❛s tr❛♥s✲ ❢♦r♠❛çõ❡s ♦❝♦rr✐❞❛s ♣❡❧❛s ♣♦❧ít✐❝❛s ❡❞✉❝❛❝✐♦♥❛✐s✱ ❡①✐❣✐♥❞♦ ❞♦ ♣r♦❢❡ss♦r ❛ ❝r✐❛çã♦ ❞❡ ❡str❛té❣✐❛s q✉❡ ❛♣r❡s❡♥t❡♠ ❛ r❡❛❧✐❞❛❞❡ ❡①tr❛❝❧❛ss❡ ❛♦ s❡✉ ❛❧✉♥♦✱ ✈✐s❛♥❞♦ ❛ ✉♠❛ ❢❛❝✐❧✐✲ t❛çã♦ ♥♦ ❞❡s❡♥r♦❧❛r ❞❡ s✐t✉❛çõ❡s q✉❡ ♣♦❞❡♠ s✉r❣✐r ❡♠ s❡✉ ❞✐❛✲❛✲❞✐❛✱ ♦ q✉❡ s❡ ✈ê ❤♦❥❡ é ✉♠ ❡♥s✐♥♦ ♣♦✉❝♦ ❝♦♥t❡①t✉❛❧✐③❛❞♦✱ ❝♦♥tr✐❜✉✐♥❞♦ ♣❛r❛ ❛ ❢❛❧t❛ ❞❡ ✐♥t❡r❡ss❡ ❞♦s ❛❧✉♥♦s✳ ◆♦ t♦❝❛♥t❡ à ♠❛t❡♠át✐❝❛✱ ✐♥❢❡❧✐③♠❡♥t❡✱ ❛ ✐❞❡✐❛ ✜❣✉r❛ ❞♦ ♠❡s♠♦ ♠♦❞♦✱ ♣♦rt❛♥t♦ tr❛❜❛❧❤❛r ❡ss❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❢♦r♠❛ ❛ ♣r❡♥❞❡r ❛ ❛t❡♥çã♦ ❞♦ ❛❧✉♥♦✱ ❛ ✜♠ ❞❡ ❡♥s✐♥á✲❧♦ ❛ ❧✐❞❛r ❝♦♠ ♦s ♠❛✐s ❞✐✈❡rs♦s ❝❛s♦s é ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛✳
◆❡ss❡ ❝♦♥t❡①t♦✱ ❛ ▼❛t❡♠át✐❝❛ ✜♥❛♥❝❡✐r❛ s❡ ❛♣r❡s❡♥t❛ ❝♦♠♦ ✉♠❛ ❡①❝❡❧❡♥t❡ ❛❧t❡r♥❛✲ t✐✈❛ ♣❛r❛ ❝♦♠♣♦r ♦ ❝✉rrí❝✉❧♦ ❡s❝♦❧❛r✱ ✈✐st♦ q✉❡ é ❝♦♥t❡①t✉❛❧ ♣♦r ❡①❝❡❧ê♥❝✐❛✱ é ❛t✉❛❧ ❡ ❞❡ ✐♠♣♦rtâ♥❝✐❛ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞❡ ✉♠ s❡r ❤✉♠❛♥♦ ❝rít✐❝♦ ❡ ♣❛r❛ ✉♠ ❜♦♠ ♣❧❛♥❡❥❛♠❡♥t♦ ❢❛♠✐❧✐❛r✱ ♣♦✐s ❡❧❛ ♦❢❡r❡❝❡ ❜❛s❡ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ t♦♠❛❞❛ ❞❡ ✐♠♣♦rt❛♥t❡s ❞❡❝✐sõ❡s ❞✉r❛♥t❡ ❛ ✈✐❞❛✳
❈♦♠♦ ❙✐t✉❛çã♦✲Pr♦❜❧❡♠❛✱ s❛❜❡✲s❡ q✉❡ ❞✐❛♥t❡ ❞❛ r❡❛❧ s✐t✉❛çã♦ ♥❛ q✉❛❧ s❡ ❡♥❝♦♥tr❛♠ ♦s ❥♦✈❡♥s ❛t✉❛❧♠❡♥t❡✱ é q✉❡ s✉r❣✐✉ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❡st✉❞❛r ❛❜♦r❞❛❣❡♥s ❝♦♥t❡①t✉❛✲ ❧✐③❛❞❛s ♣❛r❛ ❛ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ❝♦♠♦ ❢♦r♠❛ ❞❡ s❡ ♦❜t❡r ✉♠ ❜♦♠ ♣❧❛♥❡❥❛♠❡♥t♦ ❢❛♠✐❧✐❛r✳
◆❡ss❡ s❡♥t✐❞♦✱ ❛ ♣❡r❣✉♥t❛✲♣r♦❜❧❡♠❛ ❛ s❡r r❡s♣♦♥❞✐❞❛ ❛♦ ✜♥❛❧ é✿
• ◗✉❛❧ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ♣❛r❛ ♦ ♣❧❛♥❡❥❛♠❡♥t♦ ❢❛♠✐❧✐❛r ❡ ♦
♦rç❛♠❡♥t♦ ♣❡ss♦❛❧❄
❈♦♠♦ ❖❜❥❡t✐✈♦ ●❡r❛❧ ❞❡st❡ ❡st✉❞♦✱ ❜✉s❝♦✉✲s❡ ❡✈✐❞❡♥❝✐❛r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ▼❛t❡✲ ♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ♣❛r❛ ♦ ♣❧❛♥❡❥❛♠❡♥t♦ ❢❛♠✐❧✐❛r ❡ ♦ ♦rç❛♠❡♥t♦ ♣❡ss♦❛❧✳
❏á ♦s ❖❜❥❡t✐✈♦s ❊s♣❡❝í✜❝♦s sã♦✿
• ❆♥❛❧✐s❛r ❢✉♥❞❛♠❡♥t♦s✱ ❢❡rr❛♠❡♥t❛s ❡ ♠❡❧❤♦r❡s té❝♥✐❝❛s ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ❥✉r♦s ❡
▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ♥♦ ❡♥s✐♥♦ ❡s❝♦❧❛r❀
• P❡sq✉✐s❛r ❡ ❞❡s❝r❡✈❡r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦s ❝♦♥❝❡✐t♦s ❝❧áss✐❝♦s ❞❛ ▼❛t❡♠át✐❝❛ ❜ás✐❝❛
❛♣❧✐❝❛❞♦s à ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ❡ s✉❛ r❡❛❧ ♥❡❝❡ss✐❞❛❞❡ ♣❛r❛ ♦ ♣❧❛♥❡❥❛♠❡♥t♦ ❢❛♠✐❧✐❛r✳
❊st❡ ❡st✉❞♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ 3 ✭três✮ ❝❛♣ít✉❧♦s✱ ❛❧é♠ ❞❡st❛ ✐♥tr♦❞✉çã♦✱ ❝♦♥s✐❞❡r❛✲
çõ❡s ✜♥❛✐s ❡ ❛s r❡❢❡rê♥❝✐❛s✳ ❖s ❝❛♣ít✉❧♦s ❡stã♦ ❞✐✈✐❞✐❞♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
◆♦ ❝❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ♣r♦♣♦st❛ ♣❛r❛ ✐♥tr♦❞✉③✐r ♦ ❡st✉❞♦ ❡ ♦ ❝á❧❝✉❧♦ ❞❡ ❥✉r♦s ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❛tr❛✈és ❞❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳
◆♦ ❝❛♣ít✉❧♦ ✸ tr❛t❛✲s❡ ❞❛ ♣❡r❝❡♣çã♦ ✜♥❛♥❝❡✐r❛ ❞♦ s❡r ❤✉♠❛♥♦ ❡ s✉❛ ✐♥t❡r❛çã♦ ❝♦♠ ♦ ❞✐♥❤❡✐r♦✱ ❞❡✜♥✐çõ❡s ❜ás✐❝❛s ❞❡ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛✱ ❥✉r♦s s✐♠♣❧❡s✱ ❥✉r♦s ❝♦♠♣♦st♦s✱ ✉s❛♥❞♦ ❝♦♥❝❡✐t♦s ❝❧áss✐❝♦s ❞❛ ♠❛t❡♠át✐❝❛ ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦✳
❏á ♥♦ ❝❛♣ít✉❧♦ ✹✱ ❡stá ❛ ❡❞✉❝❛çã♦ ✜♥❛♥❝❡✐r❛ ✈♦❧t❛❞❛ ♣❛r❛ ❥♦✈❡♥s✱ tr❛t❛♥❞♦ ❞♦ ♦rç❛✲ ♠❡♥t♦ ❢❛♠✐❧✐❛r✱ ✐❞❡♥t✐✜❝❛çã♦ ❞♦s r❛❧♦s ♣♦r ♦♥❞❡ ❡s❝♦❛♠ ❛s ✜♥❛♥ç❛s ♣❡ss♦❛✐s✱ ❛❧é♠ ❞❛s ❜♦❛s ♣rát✐❝❛s ♣❛r❛ ❛s ✜♥❛♥❝✐❛s ♣❡ss♦❛✐s✳
◆❡st❡ ❡st✉❞♦✱ ♣r♦❝✉r♦✉✲s❡ ❡s❝r❡✈❡r ✉♠ tr❛❜❛❧❤♦ ❞❡ ❢á❝✐❧ ❡♥t❡♥❞✐♠❡♥t♦ ♣❛r❛ q✉❛❧✲ q✉❡r ♣❡ss♦❛ q✉❡ ♣♦ss✉✐ ♠❛t❡♠át✐❝❛ ❜ás✐❝❛✳
✷ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧
◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ♣r♦♣♦st❛s ❞❡ ❝♦♠♦ ✐♥tr♦❞✉③✐r ♦ ❝á❧❝✉❧♦ ❞❡ ❥✉r♦s ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞♦ ❊♥s✐♥♦ ❋✉♥✲ ❞❛♠❡♥t❛❧✱ ❛tr❛✈és ❞❡ s✐t✉❛çõ❡s ❞♦ ❝♦t✐❞✐❛♥♦✱ ♣♦r ♠❡✐♦ ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳
✷✳✶ ❆s ❢r❛çõ❡s ❡ ♦s ♥ú♠❡r♦s ❞❡❝✐♠❛✐s ♥♦ ❝á❧❝✉❧♦ ❞❛s ♣♦r❝❡♥t❛✲
❣❡♥s
P♦❞❡♠♦s ♣❡r❝❡❜❡r ♠✉✐t❛s s✐t✉❛çõ❡s ♣r❡s❡♥t❡s ♥♦ ❝♦t✐❞✐❛♥♦ ❞♦ ✐♥❞✐✈í❞✉♦ q✉❡ sã♦ ❡①✲ ♣r❡ss❛s ❡♠ ♣♦r❝❡♥t❛❣❡♠✱ q✉❡ é r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ❘❛③ã♦ ❈❡♥t❡s✐♠❛❧✱ ✉♠❛ ❢r❛çã♦ ❝✉❥♦ ❞❡♥♦♠✐♥❛❞♦r é ✶✵✵✳ P♦r ❡①❡♠♣❧♦✿
25
100 ≡ 25%
8
100 ≡ 8%
115
100 ≡ 115%
P❛r❛ ✉t✐❧✐③❛r♠♦s ♦ ❝á❧❝✉❧♦ ♠❡♥t❛❧ ❛ ✜♠ ❞❡ ❞❡t❡r♠✐♥❛r ❛ ♣♦r❝❡♥t❛❣❡♠ ❞❡ ✉♠ ✈❛❧♦r q✉❛❧q✉❡r✱ s❡ ❢❛③ ♥❡❝❡ssár✐♦ ❝♦♥❤❡❝❡r ♦s s✐❣♥✐✜❝❛❞♦s ❞❡ ❛❧❣✉♠❛s ❞❡ss❛s ♣♦r❝❡♥t❛❣❡♥s ❝♦♠♦ ✈❡♠♦s ❛ s❡❣✉✐r✿
• ✶✵✪ ❂ ❛ ❞é❝✐♠❛ ♣❛rt❡ ♦✉ ✉♠ ❞é❝✐♠♦❀
• ✷✵ ✪ ❂ ❛ q✉✐♥t❛ ♣❛rt❡ ♦✉ ✉♠ q✉✐♥t♦❀
• ✷✺✪ ❂ ❛ q✉❛rt❛ ♣❛rt❡ ♦✉ ✉♠ q✉❛rt♦❀
• ✺✵✪ ❂ ❛ ♠❡t❛❞❡ ♦✉ ✉♠ ♠❡✐♦❀
• ✼✺✪ ❂ três ✈❡③❡s ❛ q✉❛rt❛ ♣❛rt❡ ♦✉ três q✉❛rt♦s ❞♦ t♦t❛❧❀
• ✶✵✵✪ ❂ ✉♠ ✐♥t❡✐r♦ ♦✉ t✉❞♦✳
◆❡ss❡ ❝♦♥t❡①t♦ é ❢á❝✐❧ ✈❡r q✉❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✏❢r❛çã♦✑ é ✉♠ ❡❧❡♠❡♥t♦ ❜ás✐❝♦ ❢✉♥❞❛✲ ♠❡♥t❛❧ ♣❛r❛ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ❝á❧❝✉❧♦s ❞❡ ♣♦r❝❡♥t❛❣❡♠ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❞❡ ❥✉r♦s✳ P❛r❛ t❛❧ ❞❡✈❡♠♦s t❛♠❜é♠ ❡❢❡t✉❛r ❛s ❞❡✈✐❞❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❢r❛çõ❡s ❡♠ ♣♦r❝❡♥t❛✲ ❣❡♥s ❡ ❡♠ ♥ú♠❡r♦s ❞❡❝✐♠❛s ❡ ✈✐❝❡✲✈❡rs❛✳
✷✳✶✳✶ ❚r❛♥s❢♦r♠❛çõ❡s
✶✳ ❋r❛çã♦ ❡♠ ♣♦r❝❡♥t❛❣❡♠ ❡ ❡♠ ♥ú♠❡r♦ ❞❡❝✐♠❛❧
• 1 5 =
20
100 = 20% = 0,20 =
✶0,2
• 1 2 =
50
100 ≡50% = 0,50≡0,5 • 3
4 = 75
100 ≡75%≡0,75 • 1
1 = 100
100 ≡100%≡✷1
✷✳ P♦r❝❡♥t❛❣❡♠ ❡♠ ❢r❛çã♦ ❡ ❡♠ ♥ú♠❡r♦ ❞❡❝✐♠❛❧
• 30%≡ 10030 = 3
10 = 0,3 • 52%≡ 52
100 =
13
25 = 0,52 • 45%≡ 10045 = 9
20 = 0,45
✶❱✐♥t❡ ❝❡♥tés✐♠♦s ❡ ❞♦✐s ❞é❝✐♠♦s sã♦ ❡q✉✐✈❛❧❡♥t❡s✱ ❡ ❡ss❛ ♣❛ss❛❣❡♠ s❡ ❢❛③ ♥❡❝❡ssár✐❛ ❛♣❡♥❛s ♣❛r❛
❡❧✉❝✐❞❛r ❛ ♥♦t❛çã♦ ♣❛r❛ ❡st✉❞❛♥t❡s ♥ã♦ ❢❛♠✐❧✐❛r✐③❛❞♦s✳
✷❖s ✈❛❧♦r❡s ✵✱✷ ❡ ✵✱✺ ❡stã♦ ❡♠ ♥♦t❛çã♦ ✉♥✐tár✐❛ q✉❡ r❡❝❡❜❡ ❡ss❛ ♥♦♠❡♥❝❧❛t✉r❛ ♣❡❧♦ ❢❛t♦ ❞❡ ✶✵✵✪❂✶✳
• 30%≡ 10016 = 4
25 = 0,16
✸✳ ◆ú♠❡r♦ ❞❡❝✐♠❛❧ ❡♠ ❢r❛çã♦ ❡ ❡♠ ♣♦r❝❡♥t❛❣❡♠
• 0,12 = 12
100 ≡12%
• 0,37 = 37
100 ≡37%
• 0,6 = 6 10 ≡
60
100 = 60%
• 0,03 = 0,3 10 ≡
3
100 = 3%
◗✉❛♥❞♦ ♦ ♣r♦❢❡ss♦r ♦❜s❡r✈❛r q✉❡ ♦ ❛❧✉♥♦ ❝♦♠❡ç❛ ❛ ❢❛③❡r ✉s♦ ❞❡ss❡s ❝♦♥❝❡✐t♦s✱ s✉❣❡r❡✲ s❡ q✉❡ s❡❥❛♠ tr❛❜❛❧❤❛❞♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❛tr❛✈és ❞❡ ❡①❡r❝í❝✐♦s ❞✐r❡❝✐♦♥❛❞♦s ❛♦ ❝♦♥✲ t❡①t♦✳
❆s t❛①❛s ♣♦❞❡♠ s❡r ❛♣r❡s❡♥t❛❞❛s ❞❡ ❞✉❛s ❢♦r♠❛s✿
• P❡r❝❡♥t✉❛❧ ✭✪✮❀
• ❉❡❝✐♠❛❧ ♦✉ ✉♥✐tár✐❛✳
❆❜❛✐①♦✱ ♥❛ ❚❛❜❡❧❛ ✶✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ t❛①❛s ♣❡r❝❡♥t✉❛✐s ❡ ❞❡❝✐♠❛✐s ♦✉ ✉♥✐tár✐❛s✳
❚❛❜❡❧❛ ✶✿ ❆♣r❡s❡♥t❛çã♦ ❞❡ ❚❛①❛s ❚❛①❛ P❡r❝❡♥t✉❛❧ ❚❛①❛ ❉❡❝✐♠❛❧ ♦✉ ❯♥✐tár✐❛
25% 0,25
5% 0,05
1,5% 0,015
0,5% 0,005
2,5% 0,025
2% 0,02
0,18% 0,0018
1500 15
❆ s❡❣✉✐r ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s s✐t✉❛çõ❡s ♣r♦♣♦st❛s ♣❛r❛ q✉❡ t❛✐s ❝♦♥❝❡✐t♦s s❡❥❛♠ ❛♣❧✐❝❛❞♦s✳
❊①❡♠♣❧♦ ✶✳ ◆✉♠❛ ❝❧❛ss❡ ❞❡ ✸✺ ❛❧✉♥♦s✱ ✶✹ sã♦ ❤♦♠❡♥s✳ ◗✉❛❧ ❛ ♣♦r❝❡♥t❛❣❡♠ ❞❡ ♠✉✲ ❧❤❡r❡s ♥❡ss❛ ❝❧❛ss❡❄
❙♦❧✉çã♦✿
❉♦s ✸✺ ❛❧✉♥♦s ❡①✐st❡♥t❡s ♥❡ss❛ ❝❧❛ss❡✱ ✶✹ sã♦ ❤♦♠❡♥s ❡♥tã♦ ✷✶ sã♦ ♠✉❧❤❡r❡s✳ ❖ q✉❡ ♥♦s ♠♦str❛ q✉❡ ❡①✐st❡♠ ✷✶ ♠✉❧❤❡r❡s ❡♠ ✸✺ ❛❧✉♥♦s✱ ♦✉ s❡❥❛✿
21
35 = 0,6 = 60%
❖ q✉❡ ♥♦s ♠♦str❛ q✉❡ ❛ ♣♦r❝❡♥t❛❣❡♠ ❞❡ ♠✉❧❤❡r❡s ♥❡ss❛ ❝❧❛ss❡ é ❞❡ ✻✵✪✳
❊①❡♠♣❧♦ ✷✳ ❯♠ ❝❧✐❡♥t❡ ❞✐r✐❣❡✲s❡ ❛ ✉♠❛ ❧♦❥❛ ♥♦ ❞✐❛ ✷✺ ❞❡ ❥✉♥❤♦ ♣❛r❛ ❝♦♠♣r❛r ✉♠ t❡❧❡✈✐s♦r q✉❡ ❝✉st❛ ❘✩ ✶✽✽✵✱✵✵ à ✈✐st❛✳ ❈♦♠♦ ❡❧❡ só ✈❛✐ r❡❝❡❜❡r s❡✉ s❛❧ár✐♦ ♥♦ ❞✐❛ ✶✵ ❞❡ ❥✉❧❤♦ ❡❧❡ r❡s♦❧✈❡ ❝♦♠♣r❛r ❛ ♣r❛③♦✱ s♦❧✐❝✐t❛♥❞♦ q✉❡ ♦ ♣❛❣❛♠❡♥t♦ ❞❡✈❛ s❡r ♣r♦rr♦❣❛❞♦ ♣❛r❛ ♦ ❞✐❛ ❞♦ r❡❝❡❜✐♠❡♥t♦ ❞❡ s❡✉ s❛❧ár✐♦✳ ❖ ❣❡r❡♥t❡ ❞❛ ❧♦❥❛ ❛✉t♦r✐③❛ ❛ ❝♦♠♣r❛✱ ✐♥❢♦r♠❛♥❞♦ q✉❡ ♦ ❝❧✐❡♥t❡ ❞❡✈❡ ♣❛❣❛r ✻✪ ❞❡ ❥✉r♦s s♦❜r❡ ♦ ♣r❡ç♦ à ✈✐st❛ ❞♦ t❡❧❡✈✐s♦r✱ ♣❡❧♦ ♣r❛③♦ s♦❧✐❝✐t❛❞♦✳ ❉❡ss❡ ♠♦❞♦✱ q✉❛❧ ♦ ✈❛❧♦r q✉❡ ♦ ❝❧✐❡♥t❡ ❞❡✈❡rá ♣❛❣❛r ♥❛ ❞❛t❛ s♦❧✐❝✐t❛❞❛❄
❙♦❧✉çã♦✿
❈♦♠♦ s❛❜❡♠♦s q✉❡ 6% = 6
100 ❞❡ ❘✩ ✶✽✽✵✱✵✵ ❞❡✈❡♠♦s ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❛❝r❡s❝✐❞♦
♠✉❧t✐♣❧✐❝❛♥❞♦ ♦ ♣r❡ç♦ à ✈✐st❛ ♣♦r ✻ ❡ ❡♠ s❡❣✉✐❞❛ ❞✐✈✐❞✐✲❧♦ ♣♦r ✶✵✵ ❝♦♠♦ ♥✉♠❛ ♠✉❧t✐✲ ♣❧✐❝❛çã♦ ❞❡ ✉♠❛ ❢r❛çã♦ ♣♦r ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳ ❉❡ss❛ ❢♦r♠❛ t❡♠♦s✿
R$ 1880,00× 6 100 =
R$ 11280,00
100 =R$ 112,80
❱❡r✐✜❝❛♠♦s ❡♥tã♦ q✉❡ ♦ ✈❛❧♦r ❛❝r❡s❝✐❞♦ ✭❥✉r♦s✮ ❛♦ ♣r❡ç♦ à ✈✐st❛ é ❞❡ ❘✩ ✶✶✷✱✽✵ ❡ q✉❡✱ ♦ ✈❛❧♦r ❛ s❡r ♣❛❣♦ ♣♦r ❡ss❡ ❝❧✐❡♥t❡ ♥❛ ❞❛t❛ s♦❧✐❝✐t❛❞❛ é ❞❡✿
R$ 1880,00 +R$ 112,80 =R$ 1992,80.
❯♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ r❡s♦❧✈❡r ❡ss❡ ♣r♦❜❧❡♠❛✱ é ♦❜s❡r✈❛♥❞♦ q✉❡ ❛♦ ❛❝r❡s❝❡r6%s♦❜r❡
♦ ♣r❡ç♦ à ✈✐st❛ ❞♦ t❡❧❡✈✐s♦r ♦ ♥♦✈♦ ♣r❡ç♦ s❡rá ❞❡100% + 6% = 106%❞♦ ♣r❡ç♦ ❛♥t❡r✐♦r✳
❈♦♠♦ ✈✐♠♦s 106% = 106
100 = 1,06 ✭q✉❡ ❝❤❛♠❛♠♦s ❛ ❡ss❡ ♥ú♠❡r♦ ❞❡❝✐♠❛❧ ❞❡ ❢❛t♦r ❞❡
❝♦rr❡çã♦ ❞♦ ♣r❡ç♦ à ✈✐st❛✮✳ ❆ss✐♠✱ ♣❛r❛ ♦❜t❡r♠♦s ♦ ♥♦✈♦ ♣r❡ç♦✱ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r♠♦s ♦ ♣r❡ç♦ ❛♥t❡r✐♦r ♣❡❧♦ s❡✉ ❢❛t♦r ❞❡ ❝♦rr❡çã♦✳ ❉❡st❡ ♠♦❞♦✿
R$ 1880,00.1,06 =R$ 1992,80.
❊①❡♠♣❧♦ ✸✳ ❯♠ ✐♥✈❡st✐❞♦r ❝♦♠♣r♦✉ ✉♠ t❡rr❡♥♦ ♣♦r ❘✩ ✶✺✵✵✵✱✵✵ ❡ ✈❡♥❞❡✉✲♦ ✉♠ ❛♥♦ ❞❡♣♦✐s ♣♦r ❘✩ ✶✽✼✺✵✱✵✵✳ ◗✉❛❧ ♦ ❧✉❝r♦ ❡♠ ♣♦r❝❡♥t❛❣❡♠✱ q✉❡ ❡ss❡ ✐♥✈❡st✐❞♦r ♦❜t❡✈❡❄
❙♦❧✉çã♦✿
P❛r❛ ❝❛❧❝✉❧❛r♠♦s ♦ ❧✉❝r♦ ❡♠ r❡❛✐s ❞❡✈❡♠♦s ❞✐♠✐♥✉✐r ♦ ♣r❡ç♦ ❞❡ ❝♦♠♣r❛ ✭q✉❡ t❛♠✲ ❜é♠ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣r❡ç♦ ❞❡ ❝✉st♦✮ ❞♦ ♣r❡ç♦ ❞❡ ✈❡♥❞❛✱ ❛ss✐♠✿
❖ ❧✉❝r♦ ♦❜t✐❞♦ ❡♠ r❡❛✐s é R$ 18750,00−R$ 15000,00 =R$ 3750,00.
❆❣♦r❛✱ ♣❛r❛ ❝❛❧❝✉❧❛r♠♦s ♦ ❧✉❝r♦ ♦❜t✐❞♦ ❡♠ ♣♦r❝❡♥t❛❣❡♠✱ ❜❛st❛ t♦♠❛r♠♦s ♦ q✉♦❝✐✲ ❡♥t❡ ❡♥tr❡ ♦ ❧✉❝r♦ ♦❜t✐❞♦ ❡♠ r❡❛✐s ❡ ♦ ♣r❡ç♦ ❞❡ ❝✉st♦ ❞♦ t❡rr❡♥♦✿
R$ 3750,00
R$ 15000,00 = 0,25 = 25
100 = 25%
P♦rt❛♥t♦✱ ♦ ❧✉❝r♦ ♦❜t✐❞♦ ❡♠ ♣♦r❝❡♥t❛❣❡♠ é ❞❡ ✷✺✪✳
❊①❡♠♣❧♦ ✹✳ ❯♠❛ ❝♦rr❡♥t❡ ❞❡ ♦✉r♦ ❝✉❥♦ ♣r❡ç♦ ❞❡ t❛❜❡❧❛ é ❘✩ ✸✷✵✱✵✵ é ✈❡♥❞✐❞❛ ❝♦♠ ✉♠ ❞❡s❝♦♥t♦ ❞❡ ✶✺✪✳ ◗✉❛❧ ♦ ♣r❡ç♦ ❞❡ ✈❡♥❞❛❄
❙♦❧✉çã♦✿
P❛r❛ ❝❛❧❝✉❧❛r♠♦s ♦ ❞❡s❝♦♥t♦ ❡♠ r❡❛✐s ♠✉❧t✐♣❧✐❝❛♠♦s ♦ ✈❛❧♦r ❞❡ t❛❜❡❧❛ ♣❡❧❛ ❢r❛çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ✶✺✪✱ ❛ss✐♠✿
R$ 320,00× 15
100 =R$ 48,00.
❊♥tã♦ ♦ ♣r❡ç♦✱ ❡♠ r❡❛✐s ❛♣ós ♦ ❞❡s❝♦♥t♦ é✿
R$ 320,00−R$ 48,00 =R$ 272,00.
P♦❞❡rí❛♠♦s t❛♠❜é♠ ❝♦♥❝❧✉✐r q✉❡✱ s❡ ❢♦✐ ❝♦♥❝❡❞✐❞♦ 15% ❞❡ ❞❡s❝♦♥t♦✱ ❡♥tã♦ ♦ ❝♦♠✲
♣r❛❞♦r ♣❛❣♦✉100%−15% = 85%❞♦ ✈❛❧♦r ❞❡ t❛❜❡❧❛✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ♦ ❢❛t♦r ❞❡ ❝♦rr❡çã♦
✵✱✽✺ ♣❡❧♦ ♣r❡ç♦ ❞❡ t❛❜❡❧❛✱ ❡♥❝♦♥tr❛♠♦s ♦ ♣r❡ç♦ ❞❡ ✈❡♥❞❛✳ ❉❡ss❛ ❢♦r♠❛✿
R$ 320,00×0,85 =R$ 272,00.
❆ss✐♠✱ ❛ ❝♦rr❡♥t❡ ❞❡ ♦✉r♦ ❢♦✐ ✈❡♥❞✐❞❛ ♣♦r ❘✩ ✷✼✷✱✵✵✳
✷✳✷ ❆ r❡❣r❛ ❞❡ três ♥♦ ❝á❧❝✉❧♦ ❞❛s ♣♦r❝❡♥t❛❣❡♥s
❖s ❡❧❡♠❡♥t♦s ❢✉♥❞❛♠❡♥t❛✐s ♥❛s s✐t✉❛çõ❡s ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❡♠ ♣♦r❝❡♥t❛❣❡♠ sã♦✿ ♦ ✈❛❧♦r ❜ás✐❝♦✱ ❛ t❛①❛ ❞❡ ♣♦r❝❡♥t❛❣❡♠ ❡ ❛ ♣♦r❝❡♥t❛❣❡♠ ❞♦ ✈❛❧♦r ❜ás✐❝♦✳ ❖s ♣r♦❜❧❡✲ ♠❛s ♠❛✐s s✐♠♣❧❡s ❝♦♥s✐st❡♠ ❡♠ ❞❛❞♦s ❞♦✐s ❞❡ss❡s ❡❧❡♠❡♥t♦s✱ ❝❛❧❝✉❧❛r ♦ t❡r❝❡✐r♦✳ ◆❡ss❡ ♠♦♠❡♥t♦ ♦ ♣r♦❢❡ss♦r ❞❡✈❡ ♠♦t✐✈❛r ♦ ❛❧✉♥♦ ❛ ❛♣❧✐❝❛r ❝♦♥❝❡✐t♦s ❞❡ r❡❣r❛ ❞❡ três ❡ ♣r♦✲ ♣♦r❝✐♦♥❛❧✐❞❛❞❡✳ ❈❛❞❛ ❡①❡♠♣❧♦ ❞❡✈❡ s❡r ♣❡♥s❛❞♦ ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ❞❡s❡♥✈♦❧✈❛ ❡ ❛♥❛❧✐s❡ ❡st❡s ❝♦♥❝❡✐t♦s✳
❱❛♠♦s ❡♥tã♦ ❛♣r❡s❡♥t❛r ❛❧❣✉♠❛s s✐t✉❛çõ❡s ♣❛r❛ q✉❡ t❛✐s ❝♦♥❝❡✐t♦s s❡❥❛♠ ❛♣❧✐❝❛❞♦s✳
❊①❡♠♣❧♦ ✺✳ ❯♠ ❢✉♥❝✐♦♥ár✐♦✱ ❝✉❥♦ s❛❧ár✐♦ ♠❡♥s❛❧ é ❞❡ ❘✩ ✶✵✽✵✱✵✵ r❡❝❡❜❡ ✉♠ ❛✉♠❡♥t♦ ❞❡ ✺✱✷✪✳ ◗✉❛❧ é s❡✉ ♥♦✈♦ s❛❧ár✐♦❄
❙♦❧✉çã♦ ✿
P❛r❛ ❝❛❧❝✉❧❛r♠♦s ♦ ♥♦✈♦ s❛❧ár✐♦ ❞❡✈❡♠♦s ✉s❛r ♦s ❝♦♥❝❡✐t♦s ❞❛ r❡❣r❛ ❞❡ três q✉❡ ❞❡t❡r♠✐♥❛ ❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❞♦s ❡❧❡♠❡♥t♦s ❡♥✈♦❧✈✐❞♦s ❛tr❛✈és ❞❛ r❛③ã♦ ❡①✐st❡♥t❡ ❡♥tr❡ ❛s ❣r❛♥❞❡③❛s ❡♥✈♦❧✈✐❞❛s✱ ♦♥❞❡ ❳ s❡rá ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ✈❛❧♦r ❛ s❡r ❛❝r❡s❝✐❞♦ ♥♦ s❛❧ár✐♦ ❞❡ss❡ ❢✉♥❝✐♦♥ár✐♦✳ ▲♦❣♦✿
❙❛❧ár✐♦ P♦r❝❡♥t❛❣❡♠ ✭✪✮
R$ 1080,00 100
x 5,2
100×x = R$ 1080,00×5,2
x = R$ 56,16.
P♦rt❛♥t♦ ♦ ♥♦✈♦ s❛❧ár✐♦ ❞❡ss❡ ❢✉♥❝✐♦♥ár✐♦ s❡rá ❞❡R$ 1080,00+R$ 56,16 =R$ 1136,16.
❆❧t❡r♥❛t✐✈❛♠❡♥t❡✱ ♣❛r❛ r❡s♦❧✈❡r♠♦s ❡ss❡ ♣r♦❜❧❡♠❛✱ ♣♦❞❡rí❛♠♦s ✉t✐❧✐③❛r ♦ ♠❡s♠♦ ♣r♦❝❡ss♦ ❞♦ ❡①❡♠♣❧♦ ✷✱ ♦✉ s❡❥❛✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣❡❧♦ ❢❛t♦r ❞❡ ❝♦rr❡çã♦ ✶✱✵✺✷✳ ❊♥tã♦ t❡r❡♠♦s✿
R$ 1080,00.1,052 =R$ 1136,16.
❊①❡♠♣❧♦ ✻✳ ❖ ❧✐tr♦ ❞❡ ❣❛s♦❧✐♥❛ s♦❢r❡✉ ✉♠ ❛✉♠❡♥t♦ ❞❡ ✶✷✪ ❡ ♣❛ss♦✉ ❛ ❝✉st❛r ❘✩ ✸✱✵✺✳ ◗✉❛❧ é ♦ ♣r❡ç♦ ❛♥t❡r✐♦r ❛♦ ❛✉♠❡♥t♦ ❞♦ ❧✐tr♦ ❞❡ ❣❛s♦❧✐♥❛❄
❙♦❧✉çã♦✿
❙❛❧ár✐♦ P♦r❝❡♥t❛❣❡♠ ✭✪✮
R$ 3,05 112
x 100
12×x = R$ 3,05×100
x = R$ 2,724.
❆ss✐♠✱ ♦❜s❡r✈❛♠♦s q✉❡ ♦ ♣r❡ç♦ ❞❛ ❣❛s♦❧✐♥❛ ❛♥t❡r✐♦r ❛♦ ❛✉♠❡♥t♦ ❞❡ ✶✷✪ é ❞❡ ❘✩ ✷✱✼✷✹✳
❊①❡♠♣❧♦ ✼✳ ❯♠❛ ❣❡❧❛❞❡✐r❛ s♦❢r❡ ✉♠ ❛✉♠❡♥t♦ ❞❡ ✷✺✪ ❡♠ s❡✉ ♣r❡ç♦✳ ❯♠ ❝❧✐❡♥t❡ s♦❧✐✲ ❝✐t❛ ❛♦ ✈❡♥❞❡❞♦r ✉♠ ❞❡s❝♦♥t♦ s♦❜r❡ ♦ ♥♦✈♦ ♣r❡ç♦✱ ❞❡ ♠♦❞♦ q✉❡ ❡❧❡ ♣❛❣✉❡ ♣❡❧❛ ❣❡❧❛❞❡✐r❛ ♦ ♣r❡ç♦ ❛♥t❡r✐♦r ❛♦ ❛✉♠❡♥t♦✳ ❙❡♥❞♦ ❛t❡♥❞✐❞♦ ♦ s❡✉ ♣❡❞✐❞♦✱ q✉❛❧ ❞❡✈❡rá s❡r ♦ ❞❡s❝♦♥t♦ ❞❛❞♦ ❛♦ ❝❧✐❡♥t❡❄
❙♦❧✉çã♦✿
❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❛♦s ❡①❡♠♣❧♦s ❛♥t❡r✐♦r❡s ❡ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛ ❣❡❧❛❞❡✐r❛ ❝✉st❡
R$ 1800,00✳ ❈♦♠ ♦ ❛✉♠❡♥t♦ ❞❡ ✷✪✱ ✈❡♠♦s q✉❡ ❛ ❣❡❧❛❞❡✐r❛ ♣❛ss❛ ❛ ❝✉st❛r R$ 2250,00✳
❖ ✈❡♥❞❡❞♦r ❞❡✈❡ ❝♦♥❝❡❞❡r ✉♠ ❞❡s❝♦♥t♦✱ ❞❡ t❛❧ ♠♦❞♦ q✉❡ ♦ ♣r❡ç♦ r❡t♦r♥❡ ❛R$ 1800,00✱
♦✉ s❡❥❛✱ ❞❡✈❡ ❝♦♥❝❡❞❡r ✉♠ ❞❡s❝♦♥t♦ ❞❡R$ 450,00✳ ❈♦♥s✐❞❡r❛♥❞♦ x ❛ ♣♦r❝❡♥t❛❣❡♠ ❞♦
❞❡s❝♦♥t♦✱ t❡♠♦s q✉❡✿
❙❛❧ár✐♦ P♦r❝❡♥t❛❣❡♠ ✭✪✮
R$ 2250,00 100
450,00 x
❉❛í✿
2250,00×x = R$ 450,00×100
x = 20%.
▲♦❣♦✱ ♦ ❞❡s❝♦♥t♦ ❞❡✈❡ s❡r ❞❡ ✷✵✪ ♣❛r❛ q✉❡ ♦ ❝❧✐❡♥t❡ ♣❛❣✉❡ ♦ ♣r❡ç♦ ❛♥t❡r✐♦r ❛♦ ❛✉♠❡♥t♦✳
❊①❡♠♣❧♦ ✽✳ ❯♠ ❝❧✐❡♥t❡ ♣❛❣❛ ❘✩ ✷✶✷✵✱✵✵ ♣♦r ✉♠ ❡♠♣rést✐♠♦ ❞❡ ❘✩ ✷✵✵✵✱✵✵ q✉❡ ❡❧❡ ❤❛✈✐❛ t♦♠❛❞♦ ♥♦ ♠ês ❛♥t❡r✐♦r✳ ◗✉❛❧ ❛ ♣♦r❝❡♥t❛❣❡♠ q✉❡ ❡❧❡ ♣❛❣♦✉ ❞❡ ❥✉r♦s❄
❙♦❧✉çã♦✿
❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❛♦s ❛♥t❡r✐♦r❡s✱ ❞❡✜♥✐♥❞♦ x ❝♦♠♦ ♦ ❛❝rés❝✐♠♦ ♣❡r❝❡♥t✉❛❧ s♦❜r❡ ♦
✈❛❧♦r ✐♥✐❝✐❛❧ ❘✩ ✷✵✵✵✱✵✵✳ ❆ss✐♠ t❡♠♦s q✉❡✿
❙❛❧ár✐♦ P♦r❝❡♥t❛❣❡♠ ✭✪✮
R$ 2000,00 100
2120,00 100 +x
❆ss✐♠✱
2000,00×(100% +x) = R$ 2120,00×100 100% +x = 106%.
x = 106%−100%. x = 6%.
❱❡r✐✜❝❛♠♦s ❡♥tã♦ q✉❡ ❢♦✐ ♣❛❣♦ ✻✪ ❞❡ ❥✉r♦s ♣❡❧♦ ❡♠♣rést✐♠♦ t♦♠❛❞♦✳
◆♦t❡♠♦s ❛✐♥❞❛ q✉❡✿
R$ 2000,00×1,06 =R$2120,00.
❊♠❜♦r❛ ❡♠ ❛❧❣✉♥s ❝❛s♦s ❛ s♦❧✉çã♦ ❞❡ ✐♠❡❞✐❛t♦ ♣❛r❡❝❡ ó❜✈✐❛✱ ♠❡s♠♦ ❝♦♠ ❝á❧❝✉❧♦s s✐♠♣❧❡s é ♥❡❝❡ssár✐❛ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❝♦rr❡t❛ ♣❛r❛ ♥ã♦ s❡r ✐♥❞✉③✐❞♦ ❛♦ ❡rr♦✱ ❞❡ss❛ ❢♦r♠❛ ♦ ♣r♦❢❡ss♦r ♣♦❞❡rá ❝♦♥❝❧✉✐r✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❝❧❛ss❡✱ q✉❡ t♦❞❛s ❛s ♠♦✈✐♠❡♥t❛✲ çõ❡s ✜♥❛♥❝❡✐r❛s sã♦ ❜❛s❡❛❞❛s ♥❛ ❡st✐♣✉❧❛çã♦ ♣ré✈✐❛ ❞❡ t❛①❛s ❞❡ ❥✉r♦s✳
❆♦ r❡❛❧✐③❛r♠♦s ✉♠ ❡♠♣rést✐♠♦ ♦✉ ✉♠❛ ❝♦♠♣r❛ ❛ ♣r❛③♦ ❛ ❢♦r♠❛ ❞❡ ♣❛❣❛♠❡♥t♦ é ❢❡✐t❛ ♣♦r ♠❡✐♦ ❞❡ ♣r❡st❛çõ❡s ♠❡♥s❛✐s ❛❝r❡s❝✐❞❛s ❞❡ ❥✉r♦s✱ ✐st♦ é✱ ♦ ✈❛❧♦r ❞❡ q✉✐t❛çã♦ ❞♦ ❡♠♣rést✐♠♦ ♦✉ ❞❛ ❝♦♠♣r❛ ❛ ♣r❛③♦ é s✉♣❡r✐♦r ❛♦ ✈❛❧♦r ✐♥✐❝✐❛❧✳ ❆ ❡ss❛ ❞✐❢❡r❡♥ç❛ ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ ❥✉r♦s✳
❏✉r♦s ♣r♦♣♦r❝✐♦♥❛ ❛♦s ❛❧✉♥♦s ❛ r❡✈✐sã♦ ❞❡ ❝♦♥❝❡✐t♦s ❝♦♠♦ ❢r❛çõ❡s✱ ♥ú♠❡r♦s ❞❡❝✐♠❛✐s ❡ ♣♦r❝❡♥t❛❣❡♥s✳ ❚❛♠❜é♠ ✈❡r❡♠♦s ❛❞✐❛♥t❡ ♥❡ss❡ tr❛❜❛❧❤♦ ♦✉tr♦s ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s q✉❡ s❡ r❡❧❛❝✐♦♥❛♠ ❝♦♠ ❥✉r♦s✳
✸ ❈♦♥❝❡✐t♦s ❈❧áss✐❝♦s ❞❛ ▼❛t❡♠át✐❝❛ ❛♣❧✐❝❛❞♦s à ▼❛✲
t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛
◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❝♦♥❝❡✐t♦s ❝❧áss✐❝♦s ❞❛ ▼❛t❡♠át✐❝❛ ♣❛r❛ ❝♦♠♣r❡❡♥✲ ❞❡r ❛ ♣❡r❝❡♣çã♦ ✜♥❛♥❝❡✐r❛ ❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ t❡ór✐❝♦ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ ❢ór♠✉❧❛s ✉s❛❞❛s ❡♠ ♠❛t❡♠át✐❝❛ ✜♥❛♥❝❡✐r❛✱ ❥✉r♦s s✐♠♣❧❡s✱ ❥✉r♦s ❝♦♠♣♦st♦s✳
➱ ♥♦tór✐♦ q✉❡ ♠✉✐t❛s ♣❡ss♦❛s s♦❢r❡♠ ✐♠♣❛❝t♦ ♥❡❣❛t✐✈♦ ♥❛s ✜♥❛♥ç❛s ♣❡❧❛ ❛✉sê♥❝✐❛ ❞❡ ♣❡r❝❡♣çã♦ ✜♥❛♥❝❡✐r❛✳ ▼✉✐t❛s ❞❡s❝♦♥❤❡❝❡♠ ❛ ❛♣❧✐❝❛çã♦ ❞♦s ❥✉r♦s✱ q✉❡ ♠✉✐t❛s ✈❡③❡s sã♦ ❛♣❧✐❝❛❞♦s ♥♦ ♠❡r❝❛❞♦ ❞❡ ❢♦r♠❛ ❛❜✉s✐✈❛✱ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ à ❡❧✐♠✐♥❛çã♦ ❡ ♥❡❣♦❝✐❛çã♦ ❞❡ ❞í✈✐❞❛s✳
✸✳✶ ❆ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛
❆ ♠❛t❡♠át✐❝❛ ❝♦♠❡r❝✐❛❧ ❡ ✜♥❛♥❝❡✐r❛ ♥ã♦ é ♥♦✈❛✳ ❙✉❛s ❛♣❧✐❝❛çõ❡s r❡♠♦♥t❛♠ ❞❡ ♣❡rí♦❞♦s ❛♥t❡r✐♦r❡s ❛ ❈r✐st♦✳ ❆ ♣ró♣r✐❛ ❇í❜❧✐❛ ❙❛❣r❛❞❛ tr❛③ r❡❢❡rê♥❝✐❛s ❞❡ ❥✉r♦s ❡ ❞❡ ❛♣❧✐❝❛çõ❡s ✜♥❛♥❝❡✐r❛s✱ ❝♦♥❢♦r♠❡ ❬✶✾❪✳
➱ ❜❛st❛♥t❡ ❛♥t✐❣♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❥✉r♦s✱ t❡♥❞♦ s✐❞♦ ❛♠♣❧❛♠❡♥t❡ ❞✐✈✉❧❣❛❞♦ ❡ ✉t✐❧✐③❛❞♦ ❛♦ ❧♦♥❣♦ ❞❛ ❍✐stór✐❛✳ ❊ss❡ ❝♦♥❝❡✐t♦ s✉r❣✐✉ ♥❛t✉r❛❧♠❡♥t❡ q✉❛♥❞♦ ♦ ❤♦♠❡♠ ♣❡r❝❡❜❡✉ ❡①✐st✐r ✉♠❛ ❡str❡✐t❛ r❡❧❛çã♦ ❡♥tr❡ ♦ ❞✐♥❤❡✐r♦ ❡ ♦ t❡♠♣♦✳ Pr♦❝❡ss♦s ❞❡ ❛❝✉♠✉❧❛çã♦ ❞❡ ❝❛♣✐t❛❧ ❡ ❛ ❞❡s✈❛❧♦r✐③❛çã♦ ❞❛ ♠♦❡❞❛ ❧❡✈❛r✐❛♠ ♥♦r♠❛❧♠❡♥t❡ ❛ ✐❞é✐❛ ❞❡ ❥✉r♦s✱ ♣♦✐s s❡ r❡❛❧✐③❛✈❛♠ ❜❛s✐❝❛♠❡♥t❡ ❞❡✈✐❞♦ ❛♦ ✈❛❧♦r t❡♠♣♦r❛❧ ❞♦ ❞✐♥❤❡✐r♦✳❬✷✶❪
❆ ♠❛t❡♠át✐❝❛ ✜♥❛♥❝❡✐r❛ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❡st✉❞❛r ♦ ✈❛❧♦r ❞♦ ❞✐♥❤❡✐r♦ ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦✱ s❡❣✉♥❞♦ ❬✸❪✳
✸✳✶✳✶ P♦r❝❡♥t❛❣❡♠
❖ ❝á❧❝✉❧♦ ❞❡ ♣♦r❝❡♥t❛❣❡♠ é ✉♠❛ ♦♣❡r❛çã♦ ❞❛s ♠❛✐s ❛♥t✐❣❛s✱ ❡♠ t❡r♠♦s ❞❡ ❝á❧❝✉❧♦s ❝♦✲ ♠❡r❝✐❛✐s ❡ ✜♥❛♥❝❡✐r♦s✳ ❆ ❡①♣r❡ssã♦ ♣♦r ❝❡♥t♦ é ✐♥❞✐❝❛❞❛ ❣❡r❛❧♠❡♥t❡ ♣♦r ♠❡✐♦ ❞♦ s✐♥❛❧ ✪✳
◗✉❛♥❞♦ s❡ ❡❢❡t✉❛ ✉♠ ❝á❧❝✉❧♦ ❞❡ ♣♦r❝❡♥t❛❣❡♠✱ ♥❛ ✈❡r❞❛❞❡ ❡❢❡t✉❛✲s❡ ✉♠ s✐♠♣❧❡s ❝á❧❝✉❧♦ ❞❡ ♣r♦♣♦rçã♦✳
❊①❡♠♣❧♦ ✾✳ ◗✉❛❧ é ❛ ❝♦♠✐ssã♦ ❞❡ ✶✵✪ s♦❜r❡ ❘✩ ✽✵✵✱✵✵❄
❙♦❧✉çã♦✿
❖ r❛❝✐♦❝í♥✐♦ q✉❡ s❡ ❞❡✈❡ ❡♠♣r❡❣❛r ♥❛ s♦❧✉çã♦ ❞❡st❡ ♣r♦❜❧❡♠❛ s❡❣✉❡✿ ❯s❛♥❞♦ ❛ ♥♦t❛çã♦ ❞❡ r❡❣r❛ ❞❡ três✱ t❡♠✲s❡✿
R$ 800,00 ✖✖✖ 100%
x ✖✖✖ 10%
❆♣❧✐❝❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❢✉♥❞❛♠❡♥t❛❧ ❞❛s ♣r♦♣♦rçõ❡s ✭♦ ♣r♦❞✉t♦ ❞♦s ♠❡✐♦s é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞♦s ❡①tr❡♠♦s✮✱ t❡r❡♠♦s q✉❡✿
x= 8000×10%
100% =R$ 80,00.
P♦rt❛♥t♦✱ ❛ ❝♦♠✐ssã♦ ❞❡ ✶✵✪ s♦❜r❡ ❘✩ ✽✵✵✱✵✵ é ❞❡ ❘✩ ✽✵✱✵✵✳
✸✳✷ ❋✉♥çã♦ ❆✜♠
❆s ❢✉♥çõ❡s ♠❛t❡♠át✐❝❛s✱ ❡♥tr❡ t❛♥t❛s ❝❛r❛❝t❡ríst✐❝❛s✱ ❡①❛♠✐♥❛♠ ❡ ❡s♣❡❝✐✜❝❛♠ t❛♥t♦ ♦s ❝á❧❝✉❧♦s ❞♦ ❞✐❛✲❛✲❞✐❛ q✉❛♥t♦ s✐t✉❛çõ❡s ❞❡ ♠❛✐♦r ❝♦♠♣❧❡①✐❞❛❞❡✱ ✐♥❝❧✉s✐✈❡ ❛ ♣❛rt✐r ❞♦ s❡✉ ♣♦♥t♦ ❞❡ ✈✐st❛✱ ❛♥❛❧✐s❛♠ ❛s r❡❧❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ❣r❛♥❞❡③❛s✳ ❆ ♠❛t❡♠át✐❝❛ ✜♥❛♥❝❡✐r❛ r❡❧❛❝✐♦♥❛ ❛s ♦♣❡r❛çõ❡s ✜♥❛♥❝❡✐r❛s ❞❡ ❝❛♣✐t❛✐s ♥♦s r❡❣✐♠❡s ❞❡ ❥✉r♦s s✐♠♣❧❡s ❡ ❥✉r♦s ❝♦♠✲ ♣♦st♦s às ❢✉♥çõ❡s✱ ❞❡ ♠♦❞♦ ❡s♣❡❝✐❛❧ ❛ ❢✉♥çã♦ ❛✜♠ ❡ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳
❉❡✜♥✐çã♦ ✶✳ ❯♠❛ ❢✉♥çã♦ f :R−→R q✉❛♥❞♦ ❡①✐st❡♠ ❝♦♥st❛♥t❡s r❡❛✐s ❛ ❡ ❜ t❛✐s q✉❡
f(x) =ax+b✱ ♣❛r❛ t♦❞♦ x∈R✱ ❝❤❛♠❛♠♦s ❢ ❞❡ ❢✉♥çã♦ ❆✜♠✳
P♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ✉♠❛ ❝❡rt❛ ❢✉♥çã♦f :R−→R❛✜♠ ♠❡s♠♦ q✉❡ ♦s ❝♦❡✜❝✐❡♥t❡s
a❡b♥ã♦ s❡❥❛♠ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❢♦r♥❡❝✐❞♦s✳ ♦ ♥ú♠❡r♦bé ♦ ✈❛❧♦r q✉❡ ❛ ❢✉♥çã♦ r❡♣r❡s❡♥t❛
q✉❛♥❞♦x= 0✱f(0) =b q✉❡✱ t❛♠❜é♠ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✈❛❧♦r ✐♥✐❝✐❛❧ ❞❡ ❢✳ ❖ ❝♦❡✜❝✐❡♥t❡
❛✱ é ❝❛❧❝✉❧❛❞♦ ❛ ♣❛rt✐r ❞❡f(x1) ❡f(x2)q✉❡ f ❛ss✉♠❡ ♥♦s ♣♦♥t♦sx1 ❡ x2✳ ▲♦❣♦✿ f(x1) =ax1+b ❡ f(x2) = ax2+b
❖ q✉❡ ♥♦s ♠♦str❛ q✉❡✿
f(x2)−f(x1) = (ax2+b)−(ax1+b)
f(x2)−f(x1) = ax2−ax1
f(x2)−f(x1) = a(x2−x1)
P♦rt❛♥t♦
a= f(x2)−f(x1) (x2−x1)
✸✳✷✳✶ ❋✉♥çã♦ ▲✐♥❡❛r
❆ ❢✉♥çã♦ ❧✐♥❡❛r é ❛ ❜❛s❡ ♠❛t❡♠át✐❝❛ ♣❛r❛ ❛ ♥♦çã♦ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❡ é ❞❡✜♥✐❞❛ ♣♦rf :R−→R✱ ❞❛❞❛ ♣♦r f(x) = ax✳
❉❡✜♥✐çã♦ ✷✳ ❯♠❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ é ✉♠❛ ❢✉♥çã♦ f : R −→ R t❛❧ q✉❡✱ ♣❛r❛ q✉❛✐s✲
q✉❡r ♥ú♠❡r♦s r❡❛✐s m ❡ x t❡♠✲s❡✿
f(mx) = m·f(x) ♦✉ f(mx) = f(x)
m , m6= 0.
❖❜s❡r✈❛♠♦s q✉❡ ♥♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ❢ é ✉♠❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❞✐r❡t❛✱ s❡♥❞♦ q✉❡✱ ♥♦ s❡❣✉♥❞♦ ❝❛s♦✱ ❢ ❛♣r❡s❡♥t❛ ✉♠❛ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ✐♥✈❡rs❛✳ ❖s t❡♦r❡♠❛s s❡❣✉✐♥t❡s✱ ❛♣r❡s❡♥t❛♠ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣❛r❛ ❛ ❢✉♥çã♦ ❧✐♥❡❛r✳
❚❡♦r❡♠❛ ✶✳ ✭❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✮✳ ❙❡❥❛ f : R −→ R ✉♠❛
❢✉♥çã♦ ❝r❡s❝❡♥t❡✳ ❙ã♦ ❡q✉✐✈❛❧❡♥t❡s ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿
✭✐✮ f(q·x) = q·f(x) ♣❛r❛ t♦❞♦q ∈Z ❡ t♦❞♦ x∈R;
✭✐✐✮ P♦♥❞♦a =f(1)✱ t❡♠✲s❡ f(x) = ax ♣❛r❛ t♦❞♦ x∈R;
✭✐✐✐✮ f(x+y) =f(x) +f(y) ♣❛r❛ q✉❛✐sq✉❡r x, y ∈R.
❆ ❞❡♠♦♥tr❛çã♦ q✉❡ ✉t✐❧✐③❛r❡♠♦s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✶❪
❉❡♠♦♥str❛çã♦✳ P❛r❛ ♠♦str❛r q✉❡ (i) ✐♠♣❧✐❝❛ ❡♠ (ii)✱ ♣r♦✈❛r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡✱
♣❛r❛ t♦❞♦ r❛❝✐♦♥❛❧t = p
q ✱(i) ✐♠♣❧✐❝❛ ❡♠f(t·x) = t·f(x) ✱ ♣❛r❛ t♦❞♦ x∈R✳ q·f(t·x) =f(q·t·x) = f(p·x) =p·f(x)
P♦rt❛♥t♦
f(t·x) =
p q
f(x) =t·f(x).
❙❡❥❛a =f(1)✳ ❈♦♠♦f(0) =f(0·0) = 0·f(0) = 0✱ ❛ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❡ f ♥♦s ❞á a=f(1)> f(0) = 0✳ ▲♦❣♦✱ a é ♣♦s✐t✐✈♦✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s
f(t) =f(t·1) =t ✳ f(1) =t·a=a·t ♣❛r❛ t♦❞♦ t∈Q✳
❆❣♦r❛ ♠♦str❛r❡♠♦s q✉❡ s❡ t❡♠ f(x) = ax ♣❛r❛ t♦❞♦x∈R✳
❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ ❡①✐st❛ ❛❧❣✉♠ ♥ú♠❡r♦ r❡❛❧x t❛❧ q✉❡ f(x)6=ax✳
❆❞♠✐t❛ q✉❡ ax < f(x)✳ ❚❡♠♦s
x < f(x) a
❚♦♠❡ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ t t❛❧ q✉❡
x < t < f(x) a
❊♥tã♦a·x < a·t < f(x)✱ ♦✉ s❡❥❛✱a·x < f(t)< f(x)✳ ❆❜s✉r❞♦✱ ♣♦✐s ❢ é ❝r❡s❝❡♥t❡✳
(ii)✐♠♣❧✐❝❛ ❡♠ (iii)✿
❈♦♠♦ ♣❛r❛ a=f(1)✱ t❡♠✲s❡ f(x) = ax✱ ♣❛r❛ q✉❛❧q✉❡r x∈R❀
f(x=y) = a(x+y)−→f(x+y) = ax+ay=f(x) +f(y) (iii)✐♠♣❧✐❝❛ ❡♠ (i)✿
f(q·x+q·y) =f(q·x) +f(q·y)✐♠♣❧✐❝❛ q✉❡f(q·x+q·y) =q·f(x) +q·f(y)✐♠♣❧✐❝❛
q✉❡f(q·x+q·y) = q·(f(x) +f(y)) ✐♠♣❧✐❝❛ q✉❡ f(q·x+q·y) =q·f(x+y)✳
❚❡♦r❡♠❛ ✷✳ ✭❈❛r❛❝t❡r✐③❛çã♦ ❞❛ ❋✉♥çã♦ ❆✜♠✮✳ ❙❡♥❞♦ f : R −→ R ✉♠❛ ❢✉♥çã♦
♠♦♥ót♦♥❛ ✐♥❥❡t✐✈❛✳ ❙❡ ♦ ❛❝rés❝✐♠♦ f(x+t)−f(x) = g(t) ♥ã♦ ❞❡♣❡♥❞❡r ❞❡ x✱ ♠❛s
❛♣❡♥❛s ❞❡ t✱ ❡♥tã♦ f é ✉♠❛ ❢✉♥çã♦ ❛✜♠✳
❆ ❞❡♠♦♥tr❛çã♦ q✉❡ ✉t✐❧✐③❛r❡♠♦s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✶❪
❉❡♠♦♥str❛çã♦✳ P❛r❛t, r ∈R:
g(t+r) = f(x+t+r)−f(x)
g(t+r) = f((x+t) +r)−f(x+t) +f(x+t)−f(x)
g(t+r) = g(t) +g(r)
P♦rt❛♥t♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✱ ♣♦♥❞♦a =g(1) ✱ t❡♠✲
s❡g(t) = a· t ♣❛r❛ t♦❞♦h∈R✳ ❖❜s❡r✈❛♠♦s q✉❡ f(x+t)−f(x) = a·h. ❋❛③❡♥❞♦ f(0)
✐❣✉❛❧ ❛b✱ t❡♠♦s q✉❡ f(t) = a·t+b✱ ♦✉ s❡❥❛✱f(x) =ax+b ♣❛r❛ t♦❞♦ x∈R.
❊①❡♠♣❧♦ ✶✵✳ ❙❡♥❞♦ ❢✭①✮ ❂ ❛①✰❜ ✉♠❛ ❢✉♥çã♦ ❛✜♠ ❡ s❡♥❞♦ ♣ ❡ q ♥ú♠❡r♦s r❡❛✐s ❞✐s✲ t✐♥t♦s✱ ♠♦str❛r q✉❡
f(p)−f(q)
p−q =a
✳
❙♦❧✉çã♦✿
❋❛③❡♥❞♦p > q ❝♦♠ p=q+h✱ t❡♠♦s q✉❡ f(q+h)−f(q) = a(q+h)
▲♦❣♦
f(p)−f(q) = a(p−q)−→ f(p)−f(q) (p−q) =a.
❊①❡♠♣❧♦ ✶✶✳ ❖ ✈❛❧♦r ❞❡ ✉♠ ❝❛rr♦ ❞❡❝r❡s❝❡ ❧✐♥❡❛r♠❡♥t❡ ❝♦♠ ♦ t❡♠♣♦✱ ❞❡✈✐❞♦ ❛♦ ❞❡s❣❛st❡✳❙❛❜❡♥❞♦✲s❡ q✉❡ ❤♦❥❡ ❡❧❡ ✈❛❧❡ ❘✩ ✷✵✳✵✵✵✱ ✵✵✱ ❡ ❞❛q✉✐ ❛ ✺ ❛♥♦s ❘✩ ✷✳✵✵✵✱✵✵✳ ◗✉❛❧ ♦ s❡✉ ✈❛❧♦r ❞❛q✉✐ ❛ ✸ ❛♥♦s❄
❙♦❧✉çã♦✿
◆❡ss❡ ❝❛s♦✱ x r❡♣r❡s❡♥t❛ ♦ t❡♠♣♦ ❞❡ ✉s♦ ❞♦ ✈❡í❝✉❧♦ ❡t t❡♠♣♦ q✉❡ r❡st❛ ♣❛r❛ ❝♦♠✲
♣❧❡t❛r ❝✐♥❝♦ ❛♥♦s ❞❡ ✉s♦✳
❙❛❜❡♥❞♦ q✉❡f(x+t)−f(x) =a·t ❢❛③❡♥❞♦x= 0 ❡t = 5✱ t❡♠♦s✿
f(0 + 5)−f(0) = a·5
▲♦❣♦✱
a=−3.600.
P♦rt❛♥t♦✱ ❢❛③❡♥❞♦ x= 3 ❡ t= 2 t❡♠♦s✿
f(3 + 2)−f(3) =−3.600·2−→f(3) = 9.200 =R$ 9.200,00.
❖ ✈❛❧♦r ❞♦ ❝❛rr♦ ❛♣ós três ❛♥♦s ❞❡ ✉s♦ é ❞❡ R$ 9.200,00.
✸✳✸ ❋✉♥çã♦ ❊①♣♦♥❡♥❝✐❛❧
❙❡♥❞♦ a ∈ R t❛❧ q✉❡ a > 0 ❡ a = 16 ✱ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❜❛s❡ a✱ f : R −→ R+✱
❞❡♥♦t❛❞❛ ♣♦rf(x) =ax q✉❡ ♦❜❡❞❡❝❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛ s❡❣✉✐r✱ ❝♦♠ x, y ∈R✳
✭✐✮ ax·ay =ax+y❀
✭✐✐✮ a1 =a❀
✭✐✐✐✮ x < y ✐♠♣❧✐❝❛ q✉❡ ax < ay✱ q✉❛♥❞♦ a > 1 ❡ x < y ✐♠♣❧✐❝❛ q✉❡ ay < ax q✉❛♥❞♦
0< a <1
❋✐❣✉r❛ ✶✿ ●rá✜❝♦ ❞❛ ❋✉♥çã♦ ❊①♣♦♥❡♥❝✐❛❧
❙❡♥❞♦ f : R −→ R ✉♠❛ ❢✉♥çã♦ ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ✭✐✮✱f(x+y) = f(x)·f(y)✱ ♣♦✲
❞❡♠♦s ❛✜r♠❛r q✉❡ ❢ ♥ã♦ s❡rá ❞❡✜♥✐❞❛ ♥♦ ✈❛❧♦r ✵✳ ❖❜s❡r✈❡ q✉❡ ♣❛r❛ ❛❧❣✉♠x0 ∈R t❛❧
q✉❡f(x0) = 0✱ ♣❛r❛ q✉❛❧q✉❡r x∈R✱ t❡♠♦s
f(x) =f(x+ (x0−x0)) =f(x0)·f(x−x0) = 0·f(x−x0) = 0
❖ q✉❡ ♥♦s ♠♦str❛ q✉❡ ❢ s❡rá ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✳
❚❡♦r❡♠❛ ✸✳ ✭❈❛r❛❝t❡r✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✮✳ ❙❡♥❞♦ ❢ ✿R−→R+ ✉♠❛ ❢✉♥çã♦
♠♦♥ót♦♥❛ ✐♥❥❡t✐✈❛✳ sã♦ ❡q✉✐✈❛❧❡♥t❡s ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿
✭✐✮ f(q·x) = f(x)q ♣❛r❛ t♦❞♦ n∈Z ❡ t♦❞♦ x∈R✳
✭✐✐✮ f(x) =ax ♣❛r❛ t♦❞♦ x∈R✱ ♦♥❞❡ a=f(1)✳
✭✐✐✐✮ f(x+y) =f(x)·f(y) ♣❛r❛ q✉❛✐sq✉❡r x, y ∈R✳
❆ ❞❡♠♦♥str❛çã♦ q✉❡ ✉t✐❧✐③❛r❡♠♦s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✶❪
❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❛♠♦s q✉❡ ♣❛r❛ t♦❞♦ r❛❝✐♦♥❛❧ t = p
q (p ∈ Z, q ∈ N) t❡♠✲s❡ f(t·x) = f(x)t✳ ❙❛❜❡✲s❡ q✉❡ q·t=p✱ ❡f(t·x)q=f(q·t·x) = f(p·x) =f(x)p✳
P♦rt❛♥t♦✱
f(tx) =f(x)pq =f(x)t.
❊♥tã♦✱ s❡♥❞♦✱ f(1) = a✱ t❡♠♦s q✉❡ f(t) =f(1·t) =f(1)t=at ♣❛r❛ t♦❞♦ t∈Q✳
❙✉♣♦♥❞♦ q✉❡ ❢ s❡❥❛ ❝r❡s❝❡♥t❡✱ ❧♦❣♦ 1 =f(0)< f(1) =a✳ ❙✉♣♦♥❞♦ q✉❡ ❡①✐st❛x∈R
t❛❧ q✉❡ f(x)6=ax.
❙✉♣♦♥❞♦ ax < f(x)✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ t t❛❧ q✉❡ ax < at< f(x) ✳ ❈♦♠♦ ❢
é ❝r❡s❝❡♥t❡✱ t❡♥❞♦ f(x) < f(t) ❡♥tã♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ t < x✳ P♦r ❛❜s✉r❞♦ ✈❡♠♦s q✉❡
(i)−→(ii)✳
❖❜s❡r✈❛♥❞♦ (ii)−→(iii)✿
f(x+y) =ax+y =ax·ay =f(x)·f(y)
P♦rt❛♥t♦✱ (iii)−→(i)✿
f(q·x+q·y) = f(q·x)·f(q·y) = aqx·aqy = (f(x))q·(f(y)) = (f(x)·f(y))q.
❚❡♦r❡♠❛ ✹✳ ✭❈❛r❛❝t❡r✐③❛çã♦ ❞❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s✮✳ ❙❡❥❛ h:R−→R+ ✉♠❛ ❢✉♥✲
çã♦ ♠♦♥ót♦♥❛ ✐♥❥❡t✐✈❛ t❛❧ q✉❡✱ ♣❛r❛x, t∈Rq✉❛✐sq✉❡r✱ ♦ ❛❝rés❝✐♠♦ r❡❧❛t✐✈♦ h(x+t)−h(x)
h(x)
♥ã♦ ❞❡♣❡♥❞❡ ❞❡ x✱ ♠❛s ❛♣❡♥❛s ❞❡ t✳
❊♥tã♦✱ s❡ b =h(0) ❡ a = h(1)
h(0)✱ t❡♠✲s❡ h(x) =b·ax ♣❛r❛ t♦❞♦ x∈R✳
❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❞♦ q✉❡ϕ(t) = h(x+t)
h(x) ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ ①✳
❙✉❜st✐t✉✐♥❞♦ f(x) = h(x)
b ✱ ♦♥❞❡ b = h(0)✱ ❝♦♠
f(x+t)
f(x) ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ x ❡✱ ❛❣♦r❛
❝♦♠ f(0) = 1✳ P♦♥❞♦ x = 0 ♥❛ r❡❧❛çã♦ ϕ(t) = f(x+t)
f(x) ✱ t❡♠♦s ϕ(t) = f(t) ♣❛r❛
q✉❛❧q✉❡r t∈R✳ ❖❜s❡r✈❡ q✉❡ f ❝✉♠♣r❡ f(x+y) =f(x)·f(y) ♣❛r❛ q✉❛❧q✉❡r x, y ∈R✳
P♦rt❛♥t♦h(x) =b·f(x) = b·ax✳
❊①❡♠♣❧♦ ✶✷✳ ❖ ✈❛❧♦r ❞❡ ✉♠ ❛✉t♦♠ó✈❡❧ ❞❡♣r❡❝✐❛ ❛ ❝❛❞❛ ❛♥♦ ❡♠ ✶✵✪ ❝♦♠ r❡❧❛çã♦ ❛♦ ❛♥♦ ❛♥t❡r✐♦r✳ ❙❡V ❢♦r ✈❛❧♦r ❞♦ ❝❛rr♦ ♥♦ ❛♥♦ ❞❛ ❝♦♠♣r❛✱ ❛♣ós ✺ ❛♥♦s q✉❛❧ s❡rá ♦ ✈❛❧♦r
❞❡ss❡ ❛✉t♦♠ó✈❡❧❄
❙♦❧✉çã♦✿
◆❡ss❡ ❝❛s♦✱ x r❡♣r❡s❡♥t❛ ♦ t❡♠♣♦ ❞❡ ✉s♦ ❞♦ ✈❡í❝✉❧♦ ❡t t❡♠♣♦ q✉❡ r❡st❛ ♣❛r❛ ❝♦♠✲
♣❧❡t❛r ❝✐♥❝♦ ❛♥♦s ❞❡ ✉s♦✳
❈♦♥s✐❞❡r❛♥❞♦ x= 0 ❡ t= 1 ✈❡♠✿
f(x+t)
f(x) =a
t −→ f(1)
f(0) =a−→ 0,9V
V =a−→a= 0,9.
P❛r❛ x= 0 ❡t = 2 t❡♠♦s✿
f(0 + 2)
f(0) =a
2 −→ 0,92V
V =a
2 −→a= 0,9.
▲♦❣♦✱ ♣❡❧❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s t❡♠♦s f(x+t)
f(x) =a
t ❡ ❙✉❜st✐✲
t✉✐♥❞♦f(0) ♣♦r V t❡♠♦s✿
f(t) =V ·0,9t−→f(5) =V ·0,95 ✳
❊①❡♠♣❧♦ ✶✸✳ ❖ ✈❛❧♦r ❞❡ ✉♠ ❜❡♠ ✐♠ó✈❡❧ ❛✉♠❡♥t❛ ✶✵✪ ❡♠ r❡❧❛çã♦ ❛♦ ❛♥♦ ❛♥t❡r✐♦r✳ ❙❡ ♦ ✈❛❧♦r ❞♦ ✐♠ó✈❡❧ ♥♦ ❛♥♦ ❞❛ ❝♦♠♣r❛ é ❞❡ ❘✩ ✸✵✵✳✵✵✵✱ ✵✵✱ ❛♣ós ✽ ❛♥♦s✱ q✉❛❧ s❡rá ♦ ✈❛❧♦r ❞❡ss❡ ✐♠ó✈❡❧❄
❙♦❧✉çã♦✿
◆❡ss❡ ❝❛s♦✱ x r❡♣r❡s❡♥t❛ ♦ t❡♠♣♦ ❞❡ ✉s♦ ❞♦ ✈❡í❝✉❧♦ ❡t t❡♠♣♦ q✉❡ r❡st❛ ♣❛r❛ ❝♦♠✲
♣❧❡t❛r ♦✐t♦ ❛♥♦s ❞❡ ✉s♦✳
❈♦♥s✐❞❡r❛♥❞♦ x= 0 ❡ t= 1 ✈❡♠✿
f(x+t)
f(x) =a
t −→ f(1)
f(0) =a−→
1,1V
V =a−→a= 1,1.
P❛r❛ x= 0 ❡t = 2 t❡♠♦s✿
f(2)
f(0) =a
2 −→ 1,1 2V
V =a
2 −→a= 1,1.
▲♦❣♦✱ ♣❡❧❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s t❡♠♦s f(x+t)
f(x) = a
t ❡ s✉❜st✐✲
t✉✐♥❞♦ ❢✭✵✮ ♣♦r ❱ t❡♠♦s✿
f(t) =V ·1,1t−→f(8) = 300.000·1,18 = 643.076,64.
P♦rt❛♥t♦✱ ❛♣ós 8❛♥♦s s❡rá ❞❡ R$ 643.076,64✳
✸✳✹ Pr♦❣r❡ssõ❡s ❆r✐t♠ét✐❝❛s
❆ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ❡♥❝♦♥tr❛ ✉♠❛ ❣r❛♥❞❡ ❢❡rr❛♠❡♥t❛ q✉❡ s❡ ❛♣❧✐❝❛ às s✐t✉❛✲ çõ❡s ❝♦rr❡❧❛t❛s à s✉❛ ❛♣❧✐❝❛çã♦ q✉❡ sã♦ ❛s ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ❛s q✉❛✐s r❡❧❛❝✐♦♥❛♠♦s ❞✐r❡t❛♠❡♥t❡ ❛♦s ❥✉r♦s s✐♠♣❧❡s✳
❯♠❛ Pr♦❣r❡ssã♦ ❆r✐t♠ét✐❝❛ é ✉♠❛ s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛ ❡♠ q✉❡ ❝❛❞❛ t❡r♠♦✱ ❛ ♣❛r✲ t✐r ❞♦ s❡❣✉♥❞♦ t❡r♠♦✱ é ♦ r❡s✉❧t❛❞♦ ❞❛ s♦♠❛ ❞♦ t❡r♠♦ ❛♥t❡r✐♦r ❝♦♠ ✉♠❛ ❝♦♥st❛♥t❡✳ ❚❛❧ ❝♦♥st❛♥t❡ é ❝❤❛♠❛❞❛ ❞❡ r❛③ã♦ ❞❛ Pr♦❣r❡ssã♦ ❆r✐t♠ét✐❝❛✱ q✉❡ r❡♣r❡s❡♥t❛r❡♠♦s ♣♦rr✳
✸✳✹✳✶ ❚❡r♠♦ ●❡r❛❧ ❞❡ ✉♠❛ Pr♦❣r❡ssã♦ ❆r✐t♠ét✐❝❛
◆✉♠❛ Pr♦❣r❡ssã♦ ❆r✐t♠ét✐❝❛ ✭P❆✮ ♦ t❡r♠♦ ❣❡r❛❧ é ❞❛❞♦ ♣♦r✿
a2 = a1+r
a3 = a2+r=a1+r+r=a1+ 2r a4 = a3+r=a1+ 2r+r =a1+ 3r
a5 = a4+r=a1+ 3r+r =a1+ 4r a6 = a5+r=a1+ 4r+r =a1+ 5r
✳✳✳
an = a1+ (n−1)r
❖ t❡r♠♦ ❣❡r❛❧ ❞❛ P✳●✳ ♣♦❞❡ s❡r t❛♠❜é♠ ❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
an=a0·qn
▲♦❣♦✱ (an) é ✉♠❛ P✳❆✳ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ tê♠ ❝♦♦r❞❡♥❛❞❛s
(1, a1),(2, a2),(3, a3)✱ ❡t❝ . . . r❡♣r❡s❡♥t❛♠ ✉♠❛ r❡t❛✱ ❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✷✳
❋✐❣✉r❛ ✷✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ Pr♦❣r❡ssã♦ ❆r✐t♠ét✐❝❛
