CONTAGEM: NÚMEROS ESPECIAIS

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼❆❚❖ ●❘❖❙❙❖ ❉❖ ❙❯▲

■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖

▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲

▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲

❈❖◆❚❆●❊▼✿ ◆Ú▼❊❘❖❙ ❊❙P❊❈■❆■❙

❆▲❉❖ ❆▲❊❳❆◆❉❘❊ ❉❊ ▼❊◆❊❩❊❙ ❩❆◆❖◆■

❈❆▼P❖ ●❘❆◆❉❊ ✲ ▼❙

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼❆❚❖ ●❘❖❙❙❖ ❉❖ ❙❯▲ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲

▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲

❈❖◆❚❆●❊▼✿ ◆Ú▼❊❘❖❙ ❊❙P❊❈■❆■❙

❆▲❉❖ ❆▲❊❳❆◆❉❘❊ ❉❊ ▼❊◆❊❩❊❙ ❩❆◆❖◆■

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢✳➟ ❉r✳ ➟ ❊❧✐s❛❜❡t❡ ❙♦✉s❛ ❋r❡✐t❛s

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✕ ■◆▼❆✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡✳

❈❛♠♣♦ ●r❛♥❞❡ ✲ ▼❙

❆❣♦st♦ ❞❡ ✷✵✶✸

❈❖◆❚❆●❊▼✿ ◆Ú▼❊❘❖❙ ❊❙P❊❈■❆■❙

❆▲❉❖ ❆▲❊❳❆◆❉❘❊ ❉❊ ▼❊◆❊❩❊❙ ❩❆◆❖◆■

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ s✉❜♠❡t✐❞♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼❛t♦ ●r♦ss♦ ❞♦ ❙✉❧✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡✳

❆♣r♦✈❛❞♦ ♣❡❧❛ ❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳ ➟ ❉r✳ ➟ ❊❧✐s❛❜❡t❡ ❙♦✉s❛ ❋r❡✐t❛s ✲ ❯❋▼❙ Pr♦❢✳ ❉r✳ ❈❧❛✉❞❡♠✐r ❆♥✐③ ✲ ❯❋▼❙

Pr♦❢✳ ❉r✳ ▼♦✐s❡✐s ❞♦s ❙❛♥t♦s ❈❡❝❝♦♥❡❧❧♦ ✲ ❯❋▼❚

❈❛♠♣♦ ●r❛♥❞❡ ✲ ▼❙

❆❣♦st♦ ❞❡ ✷✵✶✸

❉❡❞✐❝♦ ❡ss❡ tr❛❜❛❧❤♦ ❛ ♠✐♥❤❛ q✉❡r✐❞❛ ❡s♣♦s❛✱ ❉❛♥✐❡❧❡ ❆❦❡♠✐ ❖s❤✐r♦ ❩❛♥♦♥✐✱ q✉❡ ❛♥t❡s ❞❡ t✉❞♦ ❡ ❞❡ t♦❞♦s✱ s❡♠♣r❡ ❛❝r❡❞✐t♦✉ ❡ ❝♦♥✜♦✉ ❡♠ ♠✐♠ ❡ ❡♠ ♠❡✉ s✉❝❡ss♦✳

❊♣í❣r❛❢❡

❆ ♠❛t❡♠át✐❝❛ ❛♣r❡s❡♥t❛ ✐♥✈❡♥çõ❡s tã♦ s✉t✐s q✉❡ ♣♦❞❡rã♦ s❡r✈✐r ♥ã♦ só ♣❛r❛ s❛t✐s❢❛③❡r ♦s ❝✉r✐♦s♦s✱ ♠❛s t❛♠❜é♠ ♣❛r❛ ❛✉①✐❧✐❛r ❛s ❛rt❡s ❡ ♣♦✉♣❛r ♦ tr❛❜❛❧❤♦ ❞♦s ❤♦♠❡♥s✳

❘❡♥é ❉❡s❝❛rt❡s

❆●❘❆❉❊❈■▼❊◆❚❖❙

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ♣❛✐s✱ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❡ ♣❡❧♦s s❡✉s s❛❝r✐❢í❝✐♦s ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ t❡r ❛❝❡ss♦ ❛ ❜♦❛ ❡❞✉❝❛çã♦✱ ❡ ❛♣r♦✈❡✐t❛r ❛s ♦♣♦rt✉✲ ♥✐❞❛❞❡s q✉❡ t✐✈❡✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❡s♣♦s❛✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ q✉❡ t❡✈❡ ❡♥q✉❛♥t♦ ♠❡ ❞❡❞✐❝❛✈❛ ✐♥t❡❣r❛❧♠❡♥t❡ ❛♦ ❡st✉❞♦ ❡ ❛♦ tr❛❜❛❧❤♦✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ♣❡❧❛ ❞❡❞✐❝❛çã♦ ♥♦s tr❛❜❛❧❤♦s r❡❛❧✐③❛❞♦s✱ ♣❡❧♦ ❛♣♦✐♦ ❞❛❞♦ ❛ ♥ós✱ ❡ ♣❡❧♦ s❡✉ ❡s❢♦rç♦ ❞✉r❛♥t❡ t♦❞♦ ♦ ♣r♦❥❡t♦✳

❆❣r❛❞❡ç♦ ❛ ♣r♦❢❡ss♦r❛ ❊❧✐s❛❜❡t❡ ❙♦✉s❛ ❋r❡✐t❛s✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❞❡ ♠❡✉ tr❛❜❛❧❤♦✱ ♣❡❧❛s ❤♦r❛s ❞❡❞✐❝❛❞❛s ❛ ❡❧❡✱ ♣❡❧♦ ❛♣♦✐♦✱ ❝♦♠♣r❡❡♥sã♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ✐♥❝❡♥t✐✈♦ ❞❛❞♦ ❞✉r❛♥t❡ t♦❞❛ ❛ tr❛❥❡tór✐❛ ❞♦ tr❛❜❛❧❤♦✳

❆❣r❛❞❡ç♦ ❛ ♣r♦❢❡ss♦r❛ ➱❧✈✐❛ ▼✉r❡❜ ❙❛❧❧✉♠✱ ♣♦r s✉❛ ❞❡❞✐❝❛çã♦ ❛♦ ♥♦ss♦ tr❛❜❛❧❤♦ ✐♥✐❝✐❛❧ q✉❡✱ ♣♦r ♠♦t✐✈♦s ❞❡ ❢♦rç❛ ♠❛✐♦r✱ ♥ã♦ ♣♦❞❡ s❡r ❝♦♥❝❧✉✐❞♦✱ ♠❛s ♥♦ q✉❛❧ t❛♠❜é♠ ❛♣r❡♥❞✐ ♠✉✐t♦✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❝❧❛ss❡ ♣❡❧♦s ❡st✉❞♦s✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦✱ ♣❡❧❛ ❝♦❧❛❜♦r❛çã♦ ❡ ♣❡❧❛ ❛❥✉❞❛ q✉❡ t♦❞♦s ❞❡r❛♠ ✉♥s ❛♦s ♦✉tr♦s✳

❆❣r❛❞❡ç♦ ❛♦s ❝♦❧❡❣❛s ❏♦s✐❛♥❡ ❈♦❧♦♠❜♦ P❡❞r✐♥✐ ❊sq✉✐♥❝❛ ❡ ❘♦❣ér✐♦ ❊sq✉✐♥❝❛✱ ♣❡❧♦ ✈❛❧✐♦s♦ ❛✉①✐❧✐♦ q✉❡ ❞❡r❛♠ ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✳

P♦r ✜♠ ❛❣r❛❞❡ç♦ ❛♦ P❘❖❋▼❆❚✱ à ❈❆P❊❙✱ ❡ à t♦❞❛ s✉❛ ❡q✉✐♣❡ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ ❡ ♣❡❧♦ ❛✉①✐❧✐♦ ♥❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ s♦♥❤♦✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ✐♥t❡r❡ss❛♥t❡s ❞❡♥tr♦ ❞♦ ❝❛♠♣♦ ❞❛ ❈♦♥t❛❣❡♠ ❡♥✈♦❧✈❡♥❞♦ ♦s ❈♦❡✜❝✐❡♥t❡s ❇✐♥ô♠✐❛✐s✱ ◆ú♠❡r♦s ❞❡ ❙t✐r❧✐♥❣ ❡ ◆ú♠❡r♦s ❞❡ ❊✉❧❡r✳ ▼♦str❛r❡♠♦s ❛s ❢ór♠✉❧❛s ❞❡ r❡❝♦rrê♥❝✐❛ q✉❡ ❞❡t❡r♠✐♥❛♠ ❡ss❡s ♥ú♠❡r♦s ❡s♣❡❝✐❛✐s✱ ❡ ❛tr❛✈és ❞❡❧❛s✱ ❛s r❡❧❛çõ❡s q✉❡ ❝♦♥❡❝t❛♠ ❡ss❡s ♥ú♠❡r♦s ✉♥s ❛♦s ♦✉tr♦s✳ ❚❛♠❜é♠ ♠♦str❛r❡♠♦s s❡✉s ♣❛❞rõ❡s tr✐â♥❣✉❧❛r❡s ❡ ❡①❡♠♣❧♦s ❞❡ s✉❛ ❛♣❧✐❝❛çã♦✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❈♦♥t❛❣❡♠✱ ❘❡❝♦rrê♥❝✐❛✱ ◆ú♠❡r♦s ❊s♣❡❝✐❛✐s✱ P❛s❝❛❧✱ ❙t✐r❧✐♥❣✱ ❊✉❧❡r✐❛♥♦s✳

❆❜str❛❝t

❲❡ ♣r❡s❡♥t ✐♥ t❤✐s ♣❛♣❡r ✐♥t❡r❡st✐♥❣ r❡s✉❧ts ✐♥ t❤❡ ✜❡❧❞ ♦❢ ❈♦✉t✐♥❣✭❝♦♥❜✐♥❛t✐♦♥ ❡ ♣❡r♠✉t❛t✐✲ ♦♥s✮✱ ❡♥✈♦❧✈✐♥❣ t❤❡ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts✱ ❙t✐r❧✐♥❣ ◆✉♠❜❡rs ❛♥❞ ❊✉❧❡r✐❛♥ ◆✉♠❜❡rs✳ ❲❡ ❛r❡ s❤♦✇✐♥❣ t❤❡ r❡❝✉rr❡♥❝❡ ❢♦r♠✉❧❛s t❤❡ ❣✐✈❡ ✉s t❤✐s s♣❡❝✐❛❧ ♥✉♠❜❡rs✱ ❛♥❞ ❜② t❤❡♠✱ t❤❡ r❡❧❛t✐✲ ♦♥s t❤❛t ❝♦♥♥❡❝t t❤✐s ♥✉♠❜❡r t♦ ❡❛❝❤ ♦t❤❡r✳ ❲❡ ❛❧s♦ s❤♦✇✐♥❣ t❤❡✐r tr✐❛♥❣✉❧❛r ♣❛tt❡r♥ ❛♥❞ ❡①❛♠♣❧♦s ♦❢ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✳

❑❡②✇♦r❞s✿ ❈♦✉t✐♥❣✱ ❘❡❝✉rr❡♥❝❡✱ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs✱ P❛s❝❛❧✱ ❙t✐r❧✐♥❣✱ ❊✉❧❡r✐❛♥s✳

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❈♦♥❝❡✐t♦s ❡ ❘❡s✉❧t❛❞♦s ❇ás✐❝♦s ✸

✷✳✶ ■♥❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✷✳✷ ❈♦♥t❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷✳✸ ❘❡❝♦rrê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✸ ❚r✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧ ✷✻

✸✳✶ ❈♦❡✜❝✐❡♥t❡s ❇✐♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✸✳✷ ❚r✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✹ ◆ú♠❡r♦s ❞❡ ❙t✐r❧✐♥❣ ✸✸

✹✳✶ ◆ú♠❡r♦ ❞❡ ❙t✐r❧✐♥❣ ❞❡ ❙❡❣✉♥❞❛ ❊s♣é❝✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✹✳✷ ◆ú♠❡r♦s ❞❡ ❙t✐r❧✐♥❣ ❞❡ Pr✐♠❡✐r❛ ❊s♣é❝✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✺ ◆ú♠❡r♦s ❞❡ ❊✉❧❡r ✭◆ú♠❡r♦s ❊✉❧❡r✐❛♥♦s✮ ✺✷

✻ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✺✾

✶

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❆ ♠❛t❡♠át✐❝❛ ❛♣r❡s❡♥t❛✱ ♠✉✐t❛s ✈❡③❡s✱ ✐♥t❡r❡ss❛♥t❡s ❡ ✐♥❡s♣❡r❛❞❛s ❝♦✐♥❝✐❞ê♥❝✐❛s q✉❡ ♥♦s ❞❡✐①❛♠ ✐♥tr✐❣❛❞♦s ❡ ❝✉r✐♦s♦s ❛ r❡s♣❡✐t♦ ❞❡ss❛ ❝✐ê♥❝✐❛✳ ❊ss❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ ♠❛t❡♠át✐❝❛ ❛tr❛✐ ❡ ♠♦t✐✈❛ ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s ❛ ✐♥❣r❡ss❛r❡♠ ♦✉ s❡ ❛♣r♦❢✉♥❞❛r❡♠ ❡♠ s❡✉s ❡st✉❞♦s✳

◆♦ss♦ ♦❜❥❡t✐✈♦ ♥❡ss❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ♥ú♠❡r♦s ❡s♣❡❝✐❛✐s q✉❡ ❛♣❛r❡❝❡♠ ❡♠ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❣❡♠✳ ◆ú♠❡r♦s ❝♦♠♦ ♦s ❈♦❡✜❝✐❡♥t❡s ❇✐♥ô♠✐❛✐s✱ ◆ú♠❡r♦s ❞❡ ❙t✐r❧✐♥❣ ❡ ◆ú♠❡r♦s ❞❡ ❊✉❧❡r ♣r♦❞✉③❡♠ ♣❛❞rõ❡s ❝✉r✐♦s♦s ❡ ♣♦ss✉❡♠ ❝♦♥❡①õ❡s s✉r♣r❡❡♥❞❡♥t❡s✳ ❆♣❡s❛r ❞❡ s✉❛s ❞❡✜♥✐çõ❡s ♥ã♦ t❡r❡♠ ✉♠❛ ❧✐❣❛çã♦ ❡✈✐❞❡♥t❡✱ ♦❜s❡r✈❛♠♦s ✈ár✐❛s r❡❧❛çõ❡s ❝✉r✐♦s❛s ❡♥tr❡ ❡ss❡s ♥ú♠❡r♦s✳

❆ ♠♦t✐✈❛çã♦ ♣❛r❛ ❡ss❡ tr❛❜❛❧❤♦ ❡ ❛ ❡s❝♦❧❤❛ ❞❡ss❡ t❡♠❛✱ ❢♦r❛♠ ❛s ✐♥t❡r❡ss❛♥t❡s ❝♦✐♥❝✐✲ ❞ê♥❝✐❛s q✉❡ ♣✉❞❡♠♦s ❡♥❝♦♥tr❛r ❡st✉❞❛♥❞♦ ❡ss❡s ♥ú♠❡r♦s✱ s✉❛s r❡❧❛çõ❡s ❡ ❛♣❧✐❝❛çõ❡s✳ ❊ss❛s ❝♦✐♥❝✐❞ê♥❝✐❛s✱ ♣♦❞❡♠ ❛❥✉❞❛r ♦ ♣r♦❢❡ss♦r ❛ ❞❡s♣❡rt❛r ❛ ❝✉r✐♦s✐❞❛❞❡ ❞❡ s❡✉s ❛❧✉♥♦s ♣❡❧♦ t❡♠❛ ❞❛ ❝♦♥t❛❣❡♠✱ ♠♦t✐✈❛♥❞♦✲♦s ❛ ❡st✉❞❛r✱ ❡ ❛té ❛ t❡r❡♠ ❝✉r✐♦s✐❞❛❞❡ ♣♦r ♦✉tr♦s ❝❛♠♣♦s ❞❛ ♠❛✲ t❡♠át✐❝❛ ♦♥❞❡ t❛♠❜é♠ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ❡ss❡s r❡s✉❧t❛❞♦s ✐♥❡s♣❡r❛❞♦s✳

✷

❛♣❧✐❝❛r ❛t✐✈✐❞❛❞❡s s♦❜r❡ ♦s t❡♠❛s ❞❡ ❈♦♥t❛❣❡♠ ❡ ❘❡❝♦rrê♥❝✐❛✱ ♣❛r❛ q✉❡ ❞❡ss❛ ❢♦r♠❛✱ ♣♦ss❛ ❞❡s♣❡rt❛r ❛ ❝✉r✐♦s✐❞❛❞❡ ❡ ♦ ✐♥t❡r❡ss❡ ❞❡ ❛❧✉♥♦s✳

✸

❈❛♣ít✉❧♦ ✷

❈♦♥❝❡✐t♦s ❡ ❘❡s✉❧t❛❞♦s ❇ás✐❝♦s

✷✳✶ ■♥❞✉çã♦

▲❡♠❜r❛r❡♠♦s ♥❡st❛ s❡çã♦ ❞❡ ✉♠❛ ❢❡rr❛♠❡♥t❛ ✐♠♣♦rt❛♥t❡✱ ❞❡♥♦♠✐♥❛❞❛ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉✲ çã♦ ▼❛t❡♠át✐❝❛ ✭P■▼✮✱ ✉t✐❧✐③❛❞❛ ♥❛s ❞❡♠♦♥str❛çõ❡s ❞❡ ❢❛t♦s ❡♥✈♦❧✈❡♥❞♦ ♦ ❝♦♥❥✉♥t♦

N_{= {1,2,3,4,5,· · · }} ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❯s❛r❡♠♦s ❛ ♥♦t❛çã♦ _P(n) ✐♥❞✐❝❛♥❞♦ ✉♠❛ ♣r♦✲

♣r✐❡❞❛❞❡ r❡❧❛t✐✈❛ ❛♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳

❊①❡♠♣❧♦ ✶ P(n)✿

n

X

i=1

(2i−1) = 1 + 3 + 5 + 7 +· · ·+ (2n−1) =n2

✹

✈❡r✐✜❝❛r ❡st❛ ❛✜r♠❛çã♦ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n _∈N✿

n

n

X

i=1

(2i₋1)

1 1 = 12

2 1 + 3 = 4 = 22

3 1 + 3 + 5 = 9 = 32

4 1 + 3 + 5 + 7 = 16 = 42

5 1 + 3 + 5 + 7 + 9 = 25 = 52

❖❜s❡r✈❛♥❞♦ q✉❡

n

X

i=1

(2i₋1) =

n−1

X

i=1

(2i₋1) + (2n₋1)t❡st❛r❡♠♦s ❛ ❛✜r♠❛çã♦ ♣❛r❛ ♦✉tr♦s

✈❛❧♦r❡s✿

n

n

X

i=1

(2i₋1)

6 25 + 11 = 36 = 62

7 36 + 13 = 49 = 72

8 49 + 15 = 64 = 82

9 64 + 17 = 81 = 92

10 81 + 19 = 100 = 102

❙✉♣♦♥❤❛♠♦s ❛❣♦r❛✱ ❞❡ ✉♠ ♠♦❞♦ ❣❡r❛❧✱ q✉❡ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❛❞❡✐r❛ ♣❛r❛ n = k✱ ✐st♦

é✱

k

X

i=1

(2i−1) =k2_✳

❆ ♣❛rt✐r ❞❡st❛ s✉♣♦s✐çã♦ ✈❛♠♦s ❝❛❧❝✉❧❛r

k+1

X

i=1

(2i₋1)✿

k+1

X

i=1

(2i−1) =

k

X

i=1

(2i−1) + (2(k+ 1)−1) =k2

✺

P♦rt❛♥t♦ ❛ ❛✜r♠❛çã♦ t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n=k+ 1✳ ❘❡s✉♠✐♥❞♦✿

❼ ❱❡r✐✜❝❛♠♦s q✉❡ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✱ ❝♦♠❡ç❛♥❞♦ ♣❡❧♦ n= 1✳

❼ Pr♦✈❛♠♦s q✉❡✱ s❡ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛n =k ❡♥tã♦ t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n=k+ 1✳

❆ss✐♠✱ ❛ ♣❛rt✐r ❞❡n = 1✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ❛ ❛✜r♠❛çã♦ é ✈á❧✐❞❛ ♣❛r❛n = 2✳ ❆ ♣❛rt✐r

❞❡ n = 2✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ é ✈á❧✐❞❛ ♣❛r❛ n = 3 ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ ❝♦♥❝❧✉✐♠♦s q✉❡ é

✈á❧✐❞❛ ♣❛r❛ t♦❞♦n ♥❛t✉r❛❧✳

❖ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❢❛③❡r é ❞❡♥♦♠✐♥❛❞♦ Pr✐♥❝✐♣✐♦ ❞❛ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✱ q✉❡ ♥♦s ❣❛r❛♥t❡ ❛ ✈❛❧✐❞❛❞❡ ❞❛ ❛✜r♠❛çã♦ ♣❛r❛ t♦❞♦ n∈N✳

❊①❡♠♣❧♦ ✷P(n)✿

n

X

i=1

i2

= 1 + 22

+ 32

+ 42

+_{· · ·}+n2

= n(n+ 1)(2n+ 1) 6

❊♠ ♣❛❧❛✈r❛s✿ ❆ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞♦s ♥ ♣r✐♠❡✐r♦s ♥❛t✉r❛✐s é ✐❣✉❛❧ ❛ ❢r❛çã♦ ❞❡ ♥✉✲ ♠❡r❛❞♦r ✻ ❡ ❝✉❥♦ ❞❡♥♦♠✐♥❛❞♦r é ♦ ♣r♦❞✉t♦ ❞❡ três ♥ú♠❡r♦s✱ n✱ s❡✉ ❝♦♥s❡❝✉t✐✈♦ n + 1 ❡ ♦

❝♦♥s❡❝✉t✐✈♦ ❞♦ ❞♦❜r♦ ❞❡ n✱ 2n+ 1✳

P❛ss♦ ✶✿ P❛r❛ n= 1 t❡♠♦s12

= 1 = 1.2.3 6

P❛ss♦ ✷✿ ❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n=k✱ ✐st♦ é✱

k

X

i=1

i2

= 1 + 22

+ 32

+ 42

+_{· · ·}+k2

= k(k+ 1)(2k+ 1)

6 ✳

❆ ♣❛rt✐r ❞❡st❛ s✉♣♦s✐çã♦✱ ✈❛♠♦s ❝❛❧❝✉❧❛r

k+1

X

i=1

i2_{✳ ❖❜s❡r✈❛♥❞♦ q✉❡}

k+1

X

i=1

i2

=

k

X

i=1

i2

+ (k+ 1)2_✱ t❡♠♦s q✉❡

k+1

X

i=1

i2

=

k

X

i=1

i2

+ (k+ 1)2

= k(k+ 1)(2k+ 1)

6 + (k+ 1)

✻

❉❛í ♦❜t❡♠♦s

k+1

X

i=1

i2

= k(k+ 1)(2k+ 1) + 6(k+ 1)

2

6 =

(k+ 1)(k(2k+ 1) + 6(k+ 1))

6 =

(k+ 1)(2k2

+ 7k+ 6) 6

❈♦♠♦

2k2

+7k+6 = 2k2

+6k+4+k+2 = 2(k2

+3k+2)+k+2 = 2(k+1)(k+2)+k+2 = (k+2)(2(k+1)+1)

s❡❣✉❡ q✉❡

k+1

X

i=1

i2 = (k+ 1)(2k

2

+ 7k+ 6)

6 =

(k+ 1)(k+ 2)(2(k+ 1) + 1) 6

P♦rt❛♥t♦✱ ❛ ❛✜r♠❛çã♦ t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛n =k+ 1✳

◆♦✈❛♠❡♥t❡✱ ❛ ♣❛rt✐r ❞❡ n = 1✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ❛✜r♠❛çã♦ é ✈á❧✐❞❛ ♣❛r❛ n = 2✳ ❆

♣❛rt✐r ❞❡ n = 2✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ é ✈á❧✐❞❛ ♣❛r❛ n = 3 ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ ❝♦♥❝❧✉✐♠♦s

q✉❡ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ n ♥❛t✉r❛❧✳

❊♥✉♥❝✐❛♠♦s ❛ s❡❣✉✐r ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ ❡ ❞❛r❡♠♦s ♠❛✐s ❡①❡♠♣❧♦s✳

Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ ✶ ❙❡❥❛ P(n) ✉♠❛ ❛✜r♠❛çã♦ r❡❧❛t✐✈❛ ❛♦s ♥ú♠❡r♦s

♥❛t✉r❛✐s✳ ❙❡

✭✶✮ P(n) é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n= 1 ❡

✭✷✮ P(k)✈❡r❞❛❞❡✐r❛ ✐♠♣❧✐❝❛ q✉❡ P(k+ 1) t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✱ ♣❛r❛ t♦❞♦ k ≥1✱

✼

❖✉ ❞❡ ❢♦r♠❛ ♠❛✐s ❣❡r❛❧✱

Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ ✷ ❙❡❥❛ P(n) ✉♠❛ ❛✜r♠❛çã♦ r❡❧❛t✐✈❛ ❛♦s ♥ú♠❡r♦s

♥❛t✉r❛✐s✳ ❙❡

✭✶✮ P(n) é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n=n0 ❡

✭✷✮ P(k) ✈❡r❞❛❞❡✐r❛ ✐♠♣❧✐❝❛ q✉❡P(k+ 1) t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✱ ♣❛r❛ t♦❞♦ k _≥n0✱

❡♥tã♦ P(n) é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧n _≥n0✳

❊①❡♠♣❧♦ ✸ ❈♦♥s✐❞❡r❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✱ ❞❡✜♥✐❞❛ ❞❡ ❢♦r♠❛ r❡❝✉rs✐✈❛ ♣♦r F1 =

1, F2 = 1 ❡ ❛ ♣❛rt✐r ❞❡n ≥3✱Fn=Fn−1+Fn−2✳

❆ss✐♠✱

F3 =F2+F1 = 1 + 1 = 2, F4 =F3+F2 = 2 + 1 = 3, F5 =F4+F3 = 3 + 2 = 5, F6 =

F5+F4 = 5 + 3 = 8· · · ·✱

❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱♦❜t❡♥❞♦ ❛ s❡q✉ê♥❝✐❛ (Fn)✿

(1,1,2,3,5,8,13,21,34,55,89,144,_{· · · ·})

❱❛♠♦s ♣r♦✈❛r✱ ✉s❛♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✱ ❛ s❡❣✉✐♥t❡ ❛✜r♠❛çã♦✿

P❛r❛ t♦❞♦ ♥❛t✉r❛❧ n✱ ✈❛❧❡ q✉❡

n

X

i=1

✽

❆♥t❡s ❞❛ ♣r♦✈❛✱ ✈❛♠♦s ✈❡r✐✜❝❛r ❛ ❢ór♠✉❧❛ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✿

❼ P❛r❛ n = 1✱ F1 = 1 =F3−1 ❼ P❛r❛ n = 2✱ F1 +F2 = 2 =F4−1

❼ P❛r❛ n = 3✱ F1 +F2+F3 = 4 =F5−1

❼ P❛r❛ n = 4✱ F1 +F2+F3+F4 = 7 =F6−1

❉❡ ❢❛t♦✱ ❢✉♥❝✐♦♥♦✉ q✉❛♥❞♦ n= 1,2,3 ❡4✳ P❛ss❛♠♦s ❛❣♦r❛ ❛ ♣r♦✈❛✿

P❛ss♦ ✶✿ P❛r❛ n= 1✳

1

X

i=1

Fi =F1 = 1 =F3−1

P❛ss♦ ✷✿ ❙✉♣♦♥❤❛♠♦s ❛ ❢ór♠✉❧❛ ✈á❧✐❞❛ ♣❛r❛n =k✱ ✐st♦ é✱

k

X

i=1

Fi =Fk+2−1✳

❉❡✈❡♠♦s ♣r♦✈❛r q✉❡ t❛♠❜é♠ é ✈á❧✐❞❛ ♣❛r❛n=k+1✳ ❖❜s❡r✈❛♥❞♦ q✉❡

k+1

X

i=1

Fi = k

X

i=1

Fi+Fk+1✱

k+1

X

i=1

Fi =Fk+2−1 +Fk+1 =Fk+2+Fk+1−1

❈♦♠♦✱ ♣♦r ❞❡✜♥✐çã♦✱Fk+3=Fk+2+Fk+1✱ ❝♦♥❝❧✉✐♠♦s✱

k+1

X

i=1

Fi =Fk+2−1 +Fk+1 =Fk+2+Fk+1−1 =Fk+3−1

▲♦❣♦✱ ♣❡❧♦ P■▼✱ ❛ ❢ór♠✉❧❛ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧n✳

✾

Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ ✸ ❙❡❥❛ P(n) ✉♠❛ ❛✜r♠❛çã♦ r❡❧❛t✐✈❛ ❛♦s ♥ú♠❡r♦s

♥❛t✉r❛✐s✳ ❙❡

✭✶✮ P(n)é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n= 1 ❡

✭✷✮ P(n) ✈❡r❞❛❞❡✐r❛ ♣❛r❛ 1 _≤ n _≤ k ✐♠♣❧✐❝❛ q✉❡ P(k+ 1) t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✱ ❡♥tã♦

P(n)é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧ n✳

❊①❡♠♣❧♦ ✹ ❱❛♠♦s ♣r♦✈❛r q✉❡✱ ♣❛r❛ t♦❞♦n ♥❛t✉r❛❧✱ ✈❛❧❡ ❛ ❢ór♠✉❧❛ ❞✐r❡t❛ ✭❛❞♠✐rá✈❡❧✮

Fn=

1

√

5

1 +√5 2

!n

− √1

5

1₋√5 2

!n

♦♥❞❡ F1, F2, F3. . .é ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳

P❛ss♦ ✶✿ ❱❡r✐✜❝❛♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ♣❛r❛ n= 1 ❡ n= 2✳

1

√

5

1 +√5 2

!

− √1

5

1₋√5 2

!

= 1 +

√

5₋1 +√5

2√5 = 1 =F1

1

√

5

1 +√5 2

!2

− √1

5

1−√5 2

!2

=

1 + 2√5 + 5₋1 + 2√5₋5

4√5 = 1 =F2

P❛ss♦ ✷✿ ❙✉♣♦♥❤❛♠♦s q✉❡ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧ 1≤n ≤k ❛ ❢ór♠✉❧❛

Fn =

1

√

5

1 +√5 2

!n

− √1

5

1−√5 2

!n

s❡❥❛ ✈❡r❞❛❞❡✐r❛✳

❉❡✈❡♠♦s ♠♦str❛r q✉❡ Fk+1 =

1

√

5

1 +√5 2

!k+1 − √1

5

1−√5 2

✶✵

▲❡♠❜r❛♥❞♦ q✉❡ Fk+1 =Fk+Fk−1 ❡ ✉s❛♥❞♦ ❛ ❤✐♣ót❡s❡ t❡♠♦s q✉❡✱

Fk+1 =

1

√

5

1 +√5 2

!k

−√1

5

1₋√5 2

!k

+√1

5

1 +√5 2

!k−1 − √1

5

1−√5 2

!k−1

= √1

5

1 +√5 2

!k−1

1 +√5

2 + 1

!

−√1

5

1₋√5 2

!k−1

1₋√5

2 + 1

!

= _√1 5

1 +√5 2

!k−1

3 +√5 2

!

−√1

5

1₋√5 2

!k−1

3₋√5 2

!

= √1

5

1 +√5 2

!k−1

1 +√5 2

!2

−√1

5

1₋√5 2

!k−1

1₋√5 2

!2

▲♦❣♦✱ Fk+1 =

1

√

5

1 +√5 2

!k+1 − √1

5

1₋√5 2

!k+1

✳ P♦rt❛♥t♦✱ ❛ ❢ór♠✉❧❛ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ ♥❛t✉r❛❧n✳

✷✳✷ ❈♦♥t❛❣❡♠

✶✶

❡ ▼✉❧t✐♣❧✐❝❛çã♦✳

❉❡✜♥✐çã♦ ✶ ❈♦♥s✐❞❡r❡ N ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ _{In = {1,2,3, . . . , n}}✱ ♦♥❞❡

n_∈N✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❝♦♥❥✉♥t♦ _A é ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❡ q✉❡ t❡♠_n

❡❧❡♠❡♥t♦s q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ❜✐❥❡t✐✈❛f :In→A✳

◆❡st❡ ❝❛s♦✱né ú♥✐❝♦✱ é ❞❡♥♦♠✐♥❛❞♦ ♦ ♥ú♠❡r♦ ❝❛r❞✐♥❛❧ ❞❡A❡ ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦_|A_|=n✳

❯♠❛ ❢✉♥çã♦ ❜✐❥❡t✐✈❛ f :In→A é ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦ ❞❡ ❝♦♥t❛❣❡♠ ❞♦ ❝♦♥❥✉♥t♦ A✳

■♥❞✐❝❛♥❞♦ ♣♦ra1 =f(1), a2 =f(2),· · · , an=f(n)♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♦ ❝♦♥❥✉♥t♦ ❝♦♠♦

A=_{a1, a2, . . . , an}

❊①❡♠♣❧♦ ✺ ❖ ❝♦♥❥✉♥t♦ ❞❛s ✈♦❣❛✐s A=_{a, e, i, o, u_} é ✜♥✐t♦ ❡ _|A_|= 5✳

❊①❡♠♣❧♦ ✻ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s N ♥ã♦ é ✜♥✐t♦✳ ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡ _{n ∈} N

❡ ✉♠❛ ❢✉♥çã♦ f : In → N✳ ❚♦♠❛♥❞♦ k = f(1) +f(2) +. . .+f(n) + 1 t❡♠♦s q✉❡ k ∈ N ❡

k /_{∈ {}f(1), f(2), . . . f(n)_}✳ P♦rt❛♥t♦ f ♥ã♦ é s♦❜r❡❥❡t✐✈❛✳

❊①❡♠♣❧♦ ✼ ❙✉♣♦♥❤❛ q✉❡ ❡st❡❥❛♠ ❡♠ ❝❛rt❛③ ✺ ✜❧♠❡s ❡ ✷ ♣❡ç❛s ❞❡ t❡❛tr♦ ❡ q✉❡ ✈♦❝ê ♣♦ss❛ ❛ss✐st✐r ❛ ❛♣❡♥❛s ✉♠ ❞♦s ❡✈❡♥t♦s✳ ◗✉❛♥t❛s sã♦ ❛s s✉❛s ♦♣çõ❡s ❞❡ ❡s❝♦❧❤❛❄

■♥❞✐❝❛♥❞♦ ♣♦rA=_{F1, F2, F3, F4, F5}♦ ❝♦♥❥✉♥t♦ ❞❡ ✜❧♠❡s ❡ ♣♦rB ={P1, P2}♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❡ç❛s✱ s✉❛s ♦♣çõ❡s ❡st❛rã♦ ♥♦ ❝♦♥❥✉♥t♦A_∪B =_{F1, F2, F3, F4, F5, P1, P2}✳

❈❧❛r❛♠❡♥t❡ ✈♦❝ê t❡rá 5 + 2 = 7♦♣çõ❡s ❞❡ ❡s❝♦❧❤❛✳

❊st❡ ❡①❡♠♣❧♦ ♦❜❡❞❡❝❡ ♦ ♣r✐♥❝í♣✐♦ ❜ás✐❝♦ ❞❡ ❝♦♥t❛❣❡♠ ❝❤❛♠❛❞♦ Pr✐♥❝í♣✐♦ ❞❛ ❆❞✐çã♦✿ ❙❡ ✉♠❛ ❞❡❝✐sã♦A ♣♦❞❡ s❡r t♦♠❛❞❛ ❞❡m ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ❡ ♦✉tr❛ ❞❡❝✐sã♦B ♣♦❞❡ s❡r t♦♠❛❞❛

❞❡n ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ❡♥tã♦ ❡①✐st❡♠ m+n ♠❛♥❡✐r❛s ❞❡ t♦♠❛r ✉♠❛ ♦✉ ❛ ♦✉tr❛ ❞❡❝✐sã♦ ✳

✶✷

Pr✐♥❝í♣✐♦ ❞❛ ❆❞✐çã♦ ❙❡❥❛♠ A ❡ B ❝♦♥❥✉♥t♦s ✜♥✐t♦s ❝♦♠ A_∩B = _∅✳ ❙❡ _|A_| = m ❡

|B_|=n✱ ❡♥tã♦ _|A_∪B_|=m+n✳

❊①❡♠♣❧♦ ✽ ❙✉♣♦♥❤❛ q✉❡ ❡st❡❥❛♠ ❡♠ ❝❛rt❛③ ✹ ✜❧♠❡s ❡ ✷ ♣❡ç❛s ❞❡ t❡❛tr♦ ❡ q✉❡ ✈♦❝ê ♣♦ss❛ ❢❛③❡r ❞♦✐s ♣r♦❣r❛♠❛s✳ ◗✉❛♥t❛s sã♦ ❛s s✉❛s ♦♣çõ❡s ❞❡ ❡s❝♦❧❤❛❄

■♥❞✐❝❛♥❞♦ ♣♦r A = _{F1, F2, F3, F4} ♦ ❝♦♥❥✉♥t♦ ❞❡ ✜❧♠❡s ❡ ♣♦r B = {P1, P2} ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❡ç❛s✱ s✉❛s ♦♣çõ❡s ❡st❛rã♦ ♥♦ ❝♦♥❥✉♥t♦

A_×B =_{(F1, P1),(F1, P2),(F2, P1),(F2, P2,(F3, P1),(F3, P2),(F4, P1),(F4, P2)}

P♦rt❛♥t♦ ✈♦❝ê t❡rá 4.2 = 8 ♦♣çõ❡s ❞❡ ❡s❝♦❧❤❛✳

❊st❡ ❡①❡♠♣❧♦ ♦❜❡❞❡❝❡ ♦ ♦✉tr♦ ♣r✐♥❝í♣✐♦ ❜ás✐❝♦ ❞❡ ❝♦♥t❛❣❡♠ ❝❤❛♠❛❞♦ Pr✐♥❝í♣✐♦ ❞❛ ▼✉❧✲ t✐♣❧✐❝❛çã♦✿ ❙❡ ✉♠❛ ❞❡❝✐sã♦ A ♣♦❞❡ s❡r t♦♠❛❞❛ ❞❡ m ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ❡ ♦✉tr❛ ❞❡❝✐sã♦ B

♣♦❞❡ s❡r t♦♠❛❞❛ ❞❡n ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ❡♥tã♦ ❡①✐st❡♠ m.n♠❛♥❡✐r❛s ❞❡ s❡ t♦♠❛r ❛ ❞❡❝✐sã♦ A s❡❣✉✐❞❛ ❞❛ ❞❡❝✐sã♦B ✳

❖✉tr♦ ❡♥✉♥❝✐❛❞♦✱ ♦♥❞❡ A_×B ✐♥❞✐❝❛ ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞♦ ❝♦♥❥✉♥t♦ A ♣❡❧♦ ❝♦♥❥✉♥t♦ B✿

Pr✐♥❝í♣✐♦ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❙❡❥❛♠ A ❡ B ❝♦♥❥✉♥t♦s ✜♥✐t♦s✳ ❙❡ _|A_| = m ❡ _|B_| = n✱

❡♥tã♦ |A×B|=m.n✳

❖s ❞♦✐s ♣r✐♥❝í♣✐♦s ♣♦❞❡♠ s❡r ❡st❡♥❞✐❞♦s✿

✭✶✮❊①t❡♥sã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ❆❞✐çã♦✿ ❙❡❥❛♠ A1, A2, . . . Ak sã♦ ❝♦♥❥✉♥t♦s ✜♥✐t♦s✱

❞✐s❥✉♥t♦s ✷ ❛ ✷✳ ❙❡ |A1| = m1,|A2| = m2, . . . ,|Ak| = mk✱ ❡♥tã♦ |A1 ∪ A2 ∪ . . .∪Ak| =

m1+m2+. . .+mk✳

✶✸

✜♥✐t♦s ❝♦♠|A1|=m1,|A2|=m2, . . .|Ak|=mk✱ ❡♥tã♦

|A1×A2×. . .×Ak|=m1.m2. . . . .mk✳

❊①❡♠♣❧♦ ✾ ❖ ♥ú♠❡r♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ n ❡❧❡♠❡♥t♦s é ✐❣✉❛❧ ❛ 2n✳

❈♦♥s✐❞❡r❡ A = _{a1, . . . an} ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ n ❡❧❡♠❡♥t♦s✳ P❛r❛ t♦♠❛r s✉❜❝♦♥❥✉♥t♦s

❞❡ A ♣r❡❝✐s❛♠♦s ❡s❝♦❧❤❡r s❡✉s ❡❧❡♠❡♥t♦s ❞❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ A✳ ❆ss✐♠ ✱ ♣❛r❛ ❝❛❞❛

1 ≤ i ≤ n✱ ❞❡❝✐❞✐♠♦s s❡ ai ♣❡rt❡♥❝❡rá ♦✉ ♥ã♦ ❛ ❝❛❞❛ s✉❜❝♦♥❥✉♥t♦ ❞❡ A✱ ♥♦ ❝❛s♦ ❞❡ ai

♣❡rt❡♥❝❡r ❛♦ s✉❜❝♦♥❥✉♥t♦ s❡rá ❛tr✐❜✉✐❞♦ ❛ ❡st❡ ❡❧❡♠❡♥t♦ ♦ ✈❛❧♦r ✶✱ ❝❛s♦ ❝♦♥trár✐♦ ♦ ✈❛❧♦r ✵✳ ❆ss✐♠✱ ❝♦♥s✐❞❡r❛♥❞♦ t♦❞❛s ❛s ♥✲✉♣❧❛s ❝♦♠ ❡♥tr❛❞❛s ✶ ❡ ✵ ♦❜t❡r❡♠♦s t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡A✳

❆❧❣✉♥s ❡①❡♠♣❧♦s✿

(0, . . . ,0) | {z }

n

✐♥❞✐❝❛rá ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦✱ s❡♠ ❡❧❡♠❡♥t♦s✳

(1, . . . ,1) | {z }

n

✐♥❞✐❝❛rá ♦ ❝♦♥❥✉♥t♦ A✳

(1,0,1,0. . . ,0)

| {z }

n

✐♥❞✐❝❛rá ♦ ❝♦♥❥✉♥t♦ {a1, a3}✳

❈♦♠♦ sã♦ ❞✉❛s ❛s ♦♣çõ❡s ♣❛r❛ ❝❛❞❛ ❡❧❡♠❡♥t♦✱ ♦ ♥ú♠❡r♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s é ✐❣✉❛❧ ❛

2. . . . .2 | {z }

n✈❡③❡s

= 2n_✳

❊①❡♠♣❧♦ ✶✵ ◗✉❛♥t❛s ❢✉♥çõ❡s ❡①✐st❡♠ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ A ❝♦♠ n ❡❧❡♠❡♥t♦s ❡♠ ✉♠ ❝♦♥✲

❥✉♥t♦B ❝♦♠ m ❡❧❡♠❡♥t♦s❄

❈♦♥s✐❞❡r❡ f :A_→B✱ ♦♥❞❡A=_{a1, a2, . . . , an}❡ B ={b1, b2, . . . , bm}✳

❚❡♠♦s q✉❡ ❞❡❝✐❞✐r ♦s ✈❛❧♦r❡s ❞❡f(a1), f(a2), . . . , f(an)♦♥❞❡ ❝❛❞❛ ✉♠ ♣♦❞❡ s❡r t♦♠❛❞♦ ❞❡

✶✹

P♦rt❛♥t♦ t❡r❡♠♦s ✉♠ t♦t❛❧ ❞❡m.m . . . m

| {z }

n✈❡③❡s

=mn ❢✉♥çõ❡s✳

❊①❡♠♣❧♦ ✶✶ ◗✉❛♥t❛s ❢✉♥çõ❡s ✐♥❥❡t✐✈❛s ❡①✐st❡♠ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ A ❝♦♠ n ❡❧❡♠❡♥t♦s ❡♠

✉♠ ❝♦♥❥✉♥t♦B ❝♦♠ m ❡❧❡♠❡♥t♦s✱s❡♥❞♦ n≤m?

❈♦♥s✐❞❡r❡ f :A→B✱ ♦♥❞❡A={a1, a2, . . . , an}❡ B ={b1, b2, . . . , bm}✳

❚❡♠♦s q✉❡ ❞❡❝✐❞✐r ♦s ✈❛❧♦r❡s ❞❡f(a1), f(a2), . . . , f(an)✳

❊s❝♦❧❤❡♥❞♦ ♦ ✈❛❧♦r ♣❛r❛ f(a1)✱ ♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ❞❡ m ♠❛♥❡✐r❛s✱ ❝♦♠♦ ❛ ❢✉♥çã♦ ❞❡✈❡ s❡r ✐♥❥❡t✐✈❛✱ s♦❜r❛rã♦ m₋1✈❛❧♦r❡s ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞❡ f(a2)✳

❊s❝♦❧❤✐❞♦s ♦s ✈❛❧♦r❡s ❞❡ f(a1) ❡ f(a2) s♦❜r❛rã♦ m−2 ✈❛❧♦r❡s ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞❡ f(a3) ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳

P♦rt❛♥t♦✱ ❡①✐st❡♠ m(m₋1). . .(m₋(n₋1)) ❢✉♥çõ❡s ✐♥❥❡t✐✈❛s✳

❉❡✜♥✐çã♦ ✷ ❯♠❛ ❢✉♥çã♦ ❜✐❥❡t✐✈❛ f : A _→ A✱ ♦♥❞❡ A é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✱ é

❞❡♥♦♠✐♥❛❞❛ ✉♠❛ ♣❡r♠✉t❛çã♦ ❞❡A✳

◗✉❛♥❞♦ A é ✜♥✐t♦ ❝♦♠ n ❡❧❡♠❡♥t♦s✱ ✐♥❞✐❝❛♥❞♦ A =_{a1, . . . , an}✱ ✉♠❛ ♣❡r♠✉t❛çã♦ ❞❡ A

♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r



 

a1 a2 · · · an

f(a1) f(a2) · · · f(an)



 

♦✉ s✐♠♣❧❡s♠❡♥t❡ ♣♦r f(a1)f(a2)· · ·f(an)✳

❊①❡♠♣❧♦ ✶✷ ❈♦♥s✐❞❡r❡ A = {1,2,3}✳ ❈♦♠♦ |A| = 3✱ t❡♠♦s ✉♠ t♦t❛❧ ❞❡ 3.2.1 = 6

❜✐❥❡çõ❡s✱ ❛ s❛❜❡r

✶✺

❈♦♥s✐❞❡r❛♥❞♦ ✉♠ ❝♦♥❥✉♥t♦ A = _{a1, . . . , an}✱ ❝❛❞❛ ❜✐❥❡çã♦ f : A → A ❡st❛❜❡❧❡❝❡ ✉♠❛

♦r❞❡♥❛çã♦ ❞❡ s❡✉s ❡❧❡♠❡♥t♦s✳ ❉❡st❡ ♠♦❞♦✱ t♦❞❛s ❛s s❡❣✉✐♥t❡s ♣❡r❣✉♥t❛s ✿

❼ ❉❛❞♦s n ♦❜❥❡t♦s a1, . . . an✱ ❞❡ q✉❛♥t♦s ♠♦❞♦s ♣♦❞❡♠♦s ♦r❞❡♥á✲❧♦s❄

❼ ◗✉❛♥t❛s ❜✐❥❡çõ❡s ❡①✐st❡♠ ❞❡ A ❡♠ A❄

❼ ◗✉❛♥t❛s ♣❡r♠✉t❛çõ❡s ❞❡ A ❡①✐st❡♠❄

tê♠ ❛ ♠❡s♠❛ r❡s♣♦st❛✱ ❞❛❞❛ ♣❡❧❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿

Pr♦♣♦s✐çã♦ ✶ ❖ ♥ú♠❡r♦ ❞❡ ♣❡r♠✉t❛❝õ❡s ❞❡ A=_{a1, . . . an} é ✐❣✉❛❧ ❛

n! =n(n−1). . .1

Pr♦✈❛

❖ ♥ú♠❡r♦ ❞❡ ❢✉♥çõ❡s ✐♥❥❡t✐✈❛s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ A ❝♦♠ n ❡❧❡♠❡♥t♦s ❡♠ ✉♠ ❝♦♥❥✉♥t♦ B

❝♦♠m ❡❧❡♠❡♥t♦s✱ n≤m✱ é m(m−1). . .(m−(n−1))✳ ❈♦♠♦A={a1, . . . an}✱ ✉♠❛ ❢✉♥çã♦

f :A → A é ✐♥❥❡t✐✈❛ s❡✱ ❡ s♦♠❡♥t❡ s❡ é ❜✐❥❡t✐✈❛✳ P♦rt❛♥t♦✱ ♦ ♥ú♠❡r♦ ❞❡ ❜✐❥❡çõ❡s ❞❡ A✱ ❝♦♠ n ❡❧❡♠❡♥t♦s ❡♠ A é ✐❣✉❛❧ ❛

n(n₋1). . .(n₋(n₋1)) =n(n₋1). . .1 =n!

❉❡✜♥✐çã♦ ✸ P❡r♠✉t❛çõ❡s ❝✐r❝✉❧❛r❡s✱ t❛♠❜é♠ ❝❤❛♠❛❞❛s ❞❡ ❝✐❝❧♦s✱ ❞❡ n ❡❧❡♠❡♥t♦s sã♦

♣❡r♠✉t❛çõ❡s ❞❡st❡s ❡❧❡♠❡♥t♦s ❡♠ t♦r♥♦ ❞❡ ✉♠ ❝ír❝✉❧♦✳

❯s❛r❡♠♦s ❛ ♥♦t❛çã♦ [f(a1), . . . , f(an)] ♣❛r❛ ✐♥❞✐❝❛r ✉♠ ❝✐❝❧♦✳

✶✻

1 2 3✱ 1 3 2✱ 2 1 3✱2 3 1✱3 1 2 ❡ 3 2 1

❈♦❧♦❝❛♥❞♦ ♦s ❡❧❡♠❡♥t♦s ❛♦ r❡❞♦r ❞❡ ✉♠ ❝ír❝✉❧♦ ✈❡♠♦s q✉❡

[1 2 3] = [2 3 1] = [3 1 2]❡[1 3 2] = [2 1 3] = [3 2 1]

❆ss✐♠✱ ♦ ♥ú♠❡r♦ ❞❡ ♣❡r♠✉t❛çõ❡s ❝✐r❝✉❧❛r❡s ❞♦s ❡❧❡♠❡♥t♦s ❞❡A é ✐❣✉❛❧ ❛ ✷✳

Pr♦♣♦s✐çã♦ ✷ ❙❡❥❛ A = _{a1, a2, . . . an} ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ n ❡❧❡♠❡♥t♦s✳ ❖ ♥ú♠❡r♦ ❞❡

♣❡r♠✉t❛çõ❡s ❝✐r❝✉❧❛r❡s ❞♦s ❡❧❡♠❡♥t♦s ❞❡A é ✐❣✉❛❧ ❛

(n−1)!

Pr♦✈❛

❖ ♥ú♠❡r♦ ❞❡ ♣❡r♠✉t❛çõ❡s ❞❡ n❡❧❡♠❡♥t♦s é ✐❣✉❛❧ ❛ n!✳ ◆❡st❛ ❝♦♥t❛❣❡♠ ❝❛❞❛ ♣❡r♠✉t❛çã♦

❝✐r❝✉❧❛r ❢♦✐ ❝♦♥t❛❞❛ n ✈❡③❡s✱ ♣♦rt❛♥t♦ ♦ ♥ú♠❡r♦ ❞❡ ♣❡r♠✉t❛çõ❡s ❝✐r❝✉❧❛r❡s é ✐❣✉❛❧ ❛

n!

n = (n−1)!

✶✼

Pr♦♣♦s✐çã♦ ✸ ❙❡❥❛ A = _{a1, a2, . . . an} ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ n ❡❧❡♠❡♥t♦s✳ ❖ ♥ú♠❡r♦ ❞❡

s✉❜❝♦♥❥✉♥t♦s ❞❡A ❝♦♥t❡♥❞♦ r ❡❧❡♠❡♥t♦s✱r _≤n✱ é ❡①❛t❛♠❡♥t❡

n!

r!(n−r)!

Pr♦✈❛

■♥❞✐❝❛♥❞♦ ♣♦r{b1, b2,· · · , br}✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡A❝♦♠r❡❧❡♠❡♥t♦s✱ t❡♠♦s ♣❛r❛ ❛ ❡s❝♦❧❤❛

❞❡b1 ✉♠ t♦t❛❧ ❞❡ n ♣♦ss✐❜❧✐❞❛❞❡s✳

❊♠ s❡❣✉✐❞❛✱ ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞❡b2 t❡♠♦s n−1 ♣♦ss✐❜✐❧✐❞❛❞❡s✳ ❊s❝♦❧❤✐❞♦sb1 ❡ b2✱ s♦❜r❛rã♦

n₋2 ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞❡ a3 ❡ ❛s✐♠ ♣♦r ❞✐❛♥t❡✳ ❚❡r❡♠♦s ❡♥tã♦ ✉♠ t♦t❛❧ ❞❡

n(n ₋1). . .(n ₋(r ₋1)) ❡s❝♦❧❤❛s✳ ❖❜s❡r✈❛♥❞♦ q✉❡✱ ✉♠❛ ♣❡r♠✉t❛çã♦ ❞❡ _{b1, b2,· · · , br}

❢♦r♠❛ ♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦✱ ❝❛❞❛ s✉❜❝♦♥❥✉♥t♦ ❢♦✐ ❝♦♥t❛❞♦ r! ✈❡③❡s✳ P♦rt❛♥t♦✱ ♦ ♥ú♠❡r♦ t♦t❛❧

❞❡ s✉❝♦♥❥✉♥t♦s ❝♦♠ k ❡❧❡♠❡♥t♦s é ❡①❛t❛♠❡♥t❡

n(n₋1). . .(n₋(r₋1))

r!

❡

n(n−1). . .(n−(r−1))

r! =

n(n−1). . .(n−(r−1))(n−r). . .1

r!(n₋r)! =

n!

r!(n₋r)!

❉❡✜♥✐çã♦ ✹ P❛r❛ n, r∈N_{∪ {0}}✱ ❞❡✜♥✐♠♦s 

 

n

r



 =

  

 

n!

✶✽

♦♥❞❡ ♦❜s❡r✈❛♠♦s q✉❡ 0! := 1✳

❖ s✐♠❜♦❧♦    n r  

 é ❧✐❞♦ ❝♦♠♦✿ ❝♦♠❜✐♥❛çã♦ ❞❡ n ❡❧❡♠❡♥t♦s t♦♠❛❞♦s r ❛ r✱ ❡ é ✐❣✉❛❧ ❛♦

♥ú♠❡r♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s✱ ❝♦♥t❡♥❞♦r ❡❧❡♠❡♥t♦s✱ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠n ❡❧❡♠❡♥t♦s✳

◗✉❛♥❞♦ r ≤n t❡♠♦s q✉❡

   n r    ❂    n

n₋r





✳ ❉❡ ❢❛t♦✱ s❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦

♦✉ ♦❜s❡r✈❛♥❞♦ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s✱ ❝♦♥t❡♥❞♦r ❡❧❡♠❡♥t♦s✱ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠n

❡❧❡♠❡♥t♦s é ❡①❛t❛♠❡♥t❡ ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s✱ ❝♦♥t❡♥❞♦n−r ❡❧❡♠❡♥t♦s✱ ❞❡ ✉♠

❝♦♥❥✉♥t♦ ❝♦♠ n ❡❧❡♠❡♥t♦s✳

❈♦r♦❧ár✐♦ ✶ P❛r❛ t♦❞♦ n∈N t❡♠♦s q✉❡

   n 0   +    n 1   +. . .+    n n   = 2n

Pr♦✈❛

❇❛st❛ ♥♦t❛r q✉❡

   n 0   +    n 1   +. . .+    n n  

é ♦ ♥ú♠❡r♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞♦ ❝♦♥❥✉♥t♦

A✱ ✐♥❝❧✉✐♥❞♦ ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦✱ ♣♦rt❛♥t♦ ✐❣✉❛❧ ❛2n_✳

❉❡✜♥✐çã♦ ✺ ❆rr❛♥❥♦s ❞❡n ❡❧❡♠❡♥t♦s t♦♠❛❞♦sr❛r✱ ♦♥❞❡n, r _∈N❝♦♠ _r≤n✱ sã♦ t♦❞♦s

♦s ❣r✉♣♦s ❞❡r ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s✱ q✉❡ ❞✐❢❡r❡♠ ❡♥tr❡ s✐ t❛♠❜é♠ ♣❡❧❛ ♦r❞❡♠ ❞♦sr ❡❧❡♠❡♥t♦s

q✉❡ ❝♦♠♣õ❡♠ ♦s ❣r✉♣♦s✳

◆♦t❛çã♦✿ Ar

✶✾

P♦r ❡①❡♠♣❧♦✱ s❡A =_{1,2,3_}♦ ♥ú♠❡r♦ ❞❡ ❝♦♠❜✐♥❛çõ❡s ❞❡ ❡❧❡♠❡♥t♦s ❞❡ A t♦♠❛♠♦s ✷ ❛

✷ é ✐❣✉❛❧ ❛



 

3

2 

 =

3!

2!.1! = 3✱ ❛ s❛❜❡r ✿ {1,2},{1,3} ❡ {2,3}✳

❆❣♦r❛ ♦ ♥ú♠❡r♦ ❞❡ ❛rr❛♥❥♦s ❞♦s ✸ ❡❧❡♠❡♥t♦s t♦♠❛❞♦s ✷ ❛ ✷ é ✐❣✉❛❧ ❛ A2

3 = 6✱ ❛ s❛❜❡r✿

(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)

Pr♦♣♦s✐çã♦ ✹ ❙❡❥❛ A = _{a1, a2, . . . an} ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ n ❡❧❡♠❡♥t♦s✳ ❖ ♥ú♠❡r♦ ❞❡

❛rr❛♥❥♦s ❞❡ ❡❧❡♠❡♥t♦s ❞❡ A t♦♠❛❞♦sr ❛ r✱ r _≤n✱ é ❡①❛t❛♠❡♥t❡

n! (n₋r)!

Pr♦✈❛

■♥❞✐❝❛♥❞♦ ♣♦r (b1, b2,· · ·, br)✉♠ ❛rr❛♥❥♦ ❞❡A ❝♦♠ r ❡❧❡♠❡♥t♦s✱ t❡♠♦s ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞❡

b1 ✉♠ t♦t❛❧ ❞❡ n ♣♦ss✐❜❧✐❞❛❞❡s✳

❊♠ s❡❣✉✐❞❛✱ ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞❡b2 t❡♠♦s n−1 ♣♦ss✐❜✐❧✐❞❛❞❡s✳ ❊s❝♦❧❤✐❞♦sb1 ❡ b2✱ s♦❜r❛rã♦

n−2 ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞❡ a3 ❡ ❛s✐♠ ♣♦r ❞✐❛♥t❡✳ P♦rt❛♥t♦ ♦ ♥ú♠❡r♦ ❞❡ ❛rr❛♥❥♦s ❞❡n ❡❧❡♠❡♥t♦s t♦♠❛❞♦s r ❛ r é ✐❣✉❛❧ ❛

n(n₋1). . .(n₋(r₋1)) = n(n−1). . .(n−(r−1))(n−r). . .1 (n₋r). . .1 =

n! (n₋r)!

✷✵

✷✳✸ ❘❡❝♦rrê♥❝✐❛

❊①❡♠♣❧♦ ✶✸ ✭❆ ❚♦rr❡ ❞❡ ❍❛♥♦✐✮ ❆ t♦rr❡ ❞❡ ❍❛♥♦✐ é ✉♠ ❥♦❣♦✱ ✐♥✈❡♥t❛❞♦ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❋r❛♥❝ês ❊❞♦✉❛r❞ ▲✉❝❛s ❡♠ ✶✽✽✸✱ q✉❡ ❝♦♥s✐st❡ ❞❡✿

❼ ✉♠❛ ❜❛s❡ ♦♥❞❡ ❡stã♦ ❝♦❧♦❝❛❞❛s ✸ ❤❛st❡s ✈❡rt✐❝❛✐s A✱ B ❡ C

❼ ✉♠ ❝❡rt♦ ♥ú♠❡r♦ ❞❡ ❞✐s❝♦s ❞❡ ❞✐â♠❡tr♦s ❞✐❢❡r❡♥t❡s✱ ❢✉r❛❞♦s ♥♦ ❝❡♥tr♦✱ q✉❡ s❡rã♦ ❝♦❧♦✲ ❝❛❞♦s ♥❛s ❤❛st❡s✳

◆♦ ❝♦♠❡ç♦ ❞♦ ❥♦❣♦ ♦s ❞✐s❝♦s ❡stã♦ t♦❞♦s ❝♦❧♦❝❛❞♦s ♥❛ ❤❛st❡ A✱ ❡♠ ♦r❞❡♠ ❞❡❝r❡s❝❡♥t❡ ❞❡

t❛♠❛♥❤♦✱♦ ♠❛✐♦r ❡♠❜❛✐①♦✳

❖ ♦❜❥❡t✐✈♦ ❞♦ ❥♦❣♦ é ❛ ♠✉❞❛♥ç❛ ❞❡ t♦❞♦s ♦s ❞✐s❝♦s ❞❛ ❤❛st❡ A✱ ✉s❛♥❞♦ ❛ ❤❛st❡ B✱ ♣❛r❛ ❛

❤❛st❡C✱ ♦❜❡❞❡❝❡♥❞♦ ❛s s❡❣✉✐♥t❡s r❡❣r❛s✿

❼ P♦❞❡ s❡r ♠✉❞❛❞♦ só♠❡♥t❡ ✉♠ ❞✐s❝♦ ❞❡ ❝❛❞❛ ✈❡③✳

❼ ❯♠ ❞✐s❝♦ ♠❛✐♦r ♥✉♥❝❛ ♣♦❞❡ s❡r ❝♦❧♦❝❛❞♦ s♦❜r❡ ✉♠ ❞✐s❝♦ ♠❡♥♦r✳

P❡r❣✉♥t❛✿ ◗✉❛❧ é ♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ♠♦✈✐♠❡♥t♦s q✉❡ ❞❡✈❡♠ s❡r ❢❡✐t♦s ♣❛r❛ ❛❧❝❛♥ç❛r ♦ ♦❜❥❡t✐✈♦ ❞♦ ❥♦❣♦❄

P❛r❛ r❡s♣♦♥❞❡r ❛ ❡st❛ ♣❡r❣✉♥t❛✱ ♣r✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ✐♥tr♦❞✉③✐r ✉♠❛ ♥♦t❛çã♦✳ ❙✉♣♦✲ ♥❤❛♠♦s q✉❡ t❡♠♦s ✉♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ n ❞✐s❝♦s ❡ ✐♥❞✐❝❛♠♦s ♣♦r Tn ♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡

♠♦✈✐♠❡♥t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♠✉❞á✲❧♦s ❞❛ ❤❛st❡A ♣❛r❛ ❛ ❤❛st❡C✳

❙❡n = 1✱ ❜❛st❛ ✉♠ ♠♦✈✐♠❡♥t♦✱ ❞❡ A ♣❛r❛ C✱ ♣♦rt❛♥t♦ T1 = 1✳

❙❡ n= 2✱ t❡♠♦s ❞♦✐s ❞✐s❝♦s ♥❛ ❤❛st❡ A✱ ♦ ♠❡♥♦r s♦❜r❡ ♦ ♠❛✐♦r✳ ◆❡st❡ ❝❛s♦✱ ♠✉❞❛♠♦s ♦

✷✶

❙✉♣♦♥❤❛♠♦s n = 3✳ ◆❡st❡ ❝❛s♦ ❞❡✐①❛♠♦s ♦ ❞✐s❝♦ ♠❛✐♦r ♣❛r❛❞♦ ❡ ❝♦♠ ✸ ♠♦✈✐♠❡♥t♦s

♠✉❞❛♠♦s ♦s ❞♦✐s ♠❡♥♦r❡s ♣❛r❛ ❛ ❤❛st❡B✱ ✉s❛♥❞♦ ❛ ❤❛st❡C✳ ❊♠ s❡❣✉✐❞❛✱ ♠✉❞❛♠♦s ♦ ♠❛✐♦r

♣❛r❛ ❛ ❤❛st❡C ❡ ❝♦♠ ♠❛✐s ✸ ♠♦✈✐♠❡♥t♦s ♠✉❞❛♠♦s ♦s ❞♦✐s ♠❡♥♦r❡s ❞❡ B ♣❛r❛ C✱ ✉s❛♥❞♦ ❛ A✳ P♦rt❛♥t♦ T3 = 3 + 1 + 3 = 7✳

❙✉♣♥❤❛♠♦s n = 4✳ ❋❛③❡♠♦s ♦ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦✱ ❝♦♠♦ ♥♦ ❝❛s♦ ❛♥t❡r✐♦r✱ ❞❡✐①❛♠♦s ♦

❞✐s❝♦ ♠❛✐♦r ♣❛r❛❞♦ ❡ ❝♦♠ ✼ ♠♦✈✐♠❡♥t♦s ♠✉❞❛♠♦s ♦s ✸ ♠❡♥♦r❡s ❞❡ A ♣❛r❛ B✳ ❊♠ s❡❣✉✐❞❛

♠✉❞❛♠♦s ♦ ♠❛✐♦r ❞❡A ♣❛r❛ C ❡ ❝♦♠ ♠❛✐s ✼ ♠♦✈✐♠❡♥t♦s ♠✉❞❛♠♦s ♦s ✸ ♠❡♥♦r❡s ❞❡ B ♣❛r❛ C✱ ✉s❛♥❞♦ ❛ ❤❛st❡ A✳ P♦rt❛♥t♦ T4 = 7 + 1 + 7 = 15✳

❖❜s❡r✈❛♠♦s q✉❡✱ ❡♠ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛s r❡❣r❛s ❞♦ ❥♦❣♦✱ ♣❛r❛ q✉❡ ♦ ❞✐s❝♦ ♠❛✐♦r ♣♦ss❛ s❡r ❝♦❧♦❝❛❞♦ ♥❛ ❤❛st❡C é ♥❡❝❡ssár✐♦ ❡ s✉✜❝✐❡♥t❡ q✉❡ ♦sn₋1 ❞✐s❝♦s r❡st❛♥t❡s s❡❥❛♠ tr❛♥s❢❡r✐❞♦s

♣❛r❛ ❛ ❤❛st❡ ✐♥t❡r♠❡❞✐ár✐❛✳

❆ t❛r❡❢❛ ❞❡ ♣❛ss❛r ♦s n₋1 ❞✐s❝♦s ♠❡♥♦r❡s ❞❡ A ♣❛r❛ B✱ ✉s❛♥❞♦ ❛ C é ❡q✉✐✈❛❧❡♥t❡ à ❞❡

♣❛ss❛rn₋1❞✐s❝♦s ❞❡ A ♣❛r❛ C✱ ✉s❛♥❞♦ ❛ B✱ ♣♦rt❛♥t♦ s❡rã♦ ♥❡❝❡ssár✐♦s Tn−1 ♠♦✈✐♠❡♥t♦s✳

❯♠❛ ✈❡③✱ tr❛s❢❡r✐❞♦s ♦s ❞✐s❝♦s ♠❡♥♦r❡s ♣❛r❛ ❛B✱ ❣❛st❛✲s❡ ✉♠ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ❛ ♠✉❞❛♥ç❛

❞♦ ♠❛✐♦r ♣❛r❛ ❛ ❤❛st❡ C✳ ❋✐♥❛❧♠❡♥t❡ tr❛♥s❢❡r✐♠♦s ♦s ❞✐s❝♦s ♠❡♥♦r❡s ♣❛r❛ C✱ s♦❜r❡ ♦ ❞✐s❝♦

♠❛✐♦r✱ ♥✉♠ t♦t❛❧ ❞❡Tn−1 ♠♦✈✐♠❡♥t♦s✳ P♦rt❛♥t♦

  

 

T1 = 1

Tn=Tn−1+ 1 +Tn−1 = 2Tn−1+ 1, s❡n ≥2

❆ ❚♦rr❡ ❞❡ ❍❛♥♦✐ é ✉♠ ❡①❡♠♣❧♦ ❞♦ q✉❡ é ❝❤❛♠❛❞♦ ❞❡ r❡❝♦rrê♥❝✐❛✳ ◆♦ ❝❛s♦✱ s❛❜❡♠♦s ♦ ✈❛❧♦r ❞❡ T1 ❡ ❛ ♣❛rt✐r ❞❡ n ≥ 2✱ ♦❜t❡♠♦s ♦ ✈❛❧♦r ❞❡ Tn r❡❝♦rr❡♥❞♦ ❛♦s ❝❛s♦s ❛♥t❡r✐♦r❡s✳

❊①❡♠♣❧✐✜❝❛♥❞♦✱ ♣❛r❛ ❝❛❧❝✉❧❛rT7 ❢❛③❡♠♦s ♦ s❡❣✉✐♥t❡ ♣r♦❝❡❞✐♠❡♥t♦✿

✷✷

= 8(2T3+ 1) + 7 = 16T3+ 15 = 16(2T2+ 1) + 15 = 32T2+ 31 =

= 32(2T1+ 1) + 31 = 32.3 + 31 = 96 + 31 = 127

❆ss✐♠✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r Tn✱ ♣❛r❛ q✉❛❧q✉❡r ✈❛❧♦r ❞❡ n✱ ♠❛s ❛ t❛r❡❢❛ s❡rá ár❞✉❛ q✉❛♥❞♦n

❢♦r ❣r❛♥❞❡✳

❙❡ ❢♦r ♣♦ssí✈❡❧ ❛❝❤❛r ✉♠❛ ❢ór♠✉❧❛ ❡①♣❧í❝✐t❛ ♣❛r❛Tn✱ q✉❡ ♥ã♦ ❞❡♣❡♥❞❛ ❞♦s ❝❛s♦s ❛♥t❡r✐♦r❡s✱

❝❤❛♠❛r❡♠♦s ❡st❛ ❢ór♠✉❧❛ ❞❡ s♦❧✉çã♦ ❞❛ r❡❝♦rrê♥❝✐❛✳

◆❡st❡ ❡①❡♠♣❧♦✱ ♦❜s❡r✈❛♠♦s q✉❡ T1 = 1, T2 = 3, T3 = 7, T4 = 15, T5 = 31, T6 = 63, T7 =

127 ❡ ♣♦❞❡♠♦s ❝♦♥❥❡❝t✉r❛r q✉❡Tn= 2n−1, ♣❛r❛n≥1✳

❉❡ ❢❛t♦✱ ✉s❛♥❞♦ ♦ P■▼✱ ♣r♦✈❛r❡♠♦s ❡st❛ ❛✜r♠❛çã♦✳

P❛ss♦ ✶ ❆ ❢ór♠✉❧❛ é ✈á❧✐❞❛ ♣❛r❛ n= 1✳

P❛ss♦ ✷ ❙✉♣♦♥❤❛♠♦s ❛ ❢ór♠✉❧❛ ✈á❧✐❞❛ ♣❛r❛n=k✱ ✐st♦ é✱Tk = 2k−1✱ k≥1✳ ❚❡♠♦s q✉❡

Tk+1 = 2Tk+ 1✱ ❡ ❞❛í ✉s❛♥❞♦ ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦

Tk+1 = 2(2k−1) + 1 = 2k +1

−1

P♦rt❛♥t♦✱ ❛ ❢ór♠✉❧❛ ✈❛❧❡ ♣❛r❛ k+ 1✳

❊①❡♠♣❧♦ ✶✹ ❆ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐

     

    

F1 = 1

F2 = 1

✷✸

é ♦✉tr♦ ❡①❡♠♣❧♦ ❞❡ r❡❝♦rrê♥❝✐❛✳ Pr♦✈❛♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ✉s❛♥❞♦ ♦ P■▼✱ q✉❡

Fn=

1

√

5

1 +√5 2

!n

− √1

5

1₋√5 2

!n

❆♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r ✉♠❛ té❝♥✐❝❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ s♦❧✉çõ❡s ❞❡ ❝❡rt❛s r❡❝♦rrê♥❝✐❛s ❡ ❡♠ s❡❣✉✐❞❛ ❞❛r❡♠♦s ❝♦♠♦ ❡①❡♠♣❧♦ ♥♦✈❛♠❡♥t❡ ❛ r❡❝♦rrê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐✳

❉❡✜♥✐çã♦ ✻ ❯♠❛ r❡❝♦rrê♥❝✐❛ ♣❛r❛ ❛ s❡q✉ê♥❝✐❛ (an)n∈N é ✉♠❛ ❢ór♠✉❧❛ q✉❡ ❡①♣r❡ss❛ a_n

✉s❛♥❞♦ ♦s t❡r♠♦s ❛♥t❡r✐♦r❡s ❞❛ s❡q✉ê♥❝✐❛✱ ❛ s❛❜❡r✱ a1, . . . , an−1✱ ❛ ♣❛rt✐r ❞❡ ✉♠ ❝❡rt♦ í♥❞✐❝❡

n0 ∈N✳ ❖s ✈❛❧♦r❡s ❞❡ a0, a1, . . . , an0 sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✳

◆♦ ❡①❡♠♣❧♦ ❞❛ ❚♦rr❡ ❞❡ ❍❛♥♦✐✱ ❛ r❡❝♦rrê♥❝✐❛ éTn= 2.Tn−1+1❡ ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ❛ r❡❝♦rrê♥❝✐❛ é Fn=Fn−1+Fn−2✳

❯♠❛ s❡q✉ê♥❝✐❛(an)é ❝❤❛♠❛❞❛ s♦❧✉çã♦ ❞❛ r❡❝♦rrê♥❝✐❛ s❡ s❡✉s t❡r♠♦s s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦

❞❡ r❡❝♦rrê♥❝✐❛✳

❖❜s❡r✈❛♠♦s q✉❡ ❛ ❡q✉❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ❞❡t❡r♠✐✲ ♥❛♠ ❛ s♦❧✉çã♦ ❞❛ r❡❝♦rrê♥❝✐❛✳

❉❡✜♥✐çã♦ ✼ ❯♠❛ r❡❝♦rrê♥❝✐❛ é ❞✐t❛ ❧✐♥❡❛r ❞❡ ♦r❞❡♠ k✱ ❤♦♠♦❣ê♥❡❛ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s

❝♦♥st❛♥t❡s ❡♠ ✉♠❛ ✈❛r✐á✈❡❧ s❡ ❛ ❡q✉❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ é ❞♦ t✐♣♦

an=c1an−1+c2an−2+· · ·+ckan−k

♦♥❞❡ c1, c2, . . . , ck sã♦ ❝♦♥st❛♥t❡s r❡❛✐s✳

✷✹

Pr♦♣♦s✐çã♦ ✾ ❙❡❥❛♠c1 ❡c2 ♥ú♠❡r♦s r❡❛✐s t❛✐s q✉❡ ♦ ♣♦❧✐♥ô♠✐♦ p(x) = x2−c1x−c2 tê♠ ❞✉❛s r❛í③❡s r❡❛✐s ❞✐st✐♥t❛sr1 ❡ r2✳ ❊♥tã♦ ❛ s❡q✉ê♥❝✐❛(an) é s♦❧✉çã♦ ❞❛ r❡❝♦rrê♥❝✐❛

an =c1an−1+c2an−2

s❡✱ ❡ s♦♠❡♥t❡ s❡✱

an=α1r1n+α2rn2

♣❛r❛ n∈N_{∪ {0}}❡ ♦♥❞❡ _{α1, α2} sã♦ ❝♦♥st❛♥t❡s r❡❛✐s✳

Pr♦✈❛

❙✉♣♦♥❤❛♠♦s q✉❡ an = α1rn1 + α2rn2✱ ♦♥❞❡ r1 ❡ r2 sã♦ ❛s r❛í③❡s ❞♦ ♣♦❧✐♥ô♠✐♦ p(x) =

x2

−c1x−c2✳

❙❡❣✉❡ q✉❡

r2

1−c1r1−c2 = 0 ❡r22−c1r2−c2 = 0 ❡

c1an−1+c2an−2 =c1(α1rn− 1

1 +α2rn− 1

2 ) +c2(α1rn− 2

1 +α2rn− 2 2 ) =

=α1rn− 2

1 (c1r1+c2) +α2rn− 2

2 (c1r2+c2) =

=α1rn− 2 1 (r

2

1) +α2rn− 2 2 (r

2

2) =α1r1n+α2r2n=an

❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡(an)é s♦❧✉çã♦ ❞❛ r❡❝♦rrê♥❝✐❛ an=c1an−1+c2an−2 ❝♦♠ a0 =A ❡

a1 =B ✭❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✮✳

❱❛♠♦s ♠♦str❛r q✉❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s r❡❛✐s α1 ❡ α2 t❛✐s q✉❡ an =α1rn1 +α2rn2✱ ♦♥❞❡ r1 ❡r2 sã♦ ❛s r❛í③❡s ❞♦ ♣♦❧✐♥ô♠✐♦ p(x) = x2−c1x−c2✳

✷✺

❘❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛ ❡♥❝♦♥tr❛♠♦s

α1 =

B₋Ar2

r1−r2

α2 =

A₋Br1

r1−r2

❊①❡♠♣❧♦ ✶✺ ❱❛♠♦s ❞❡t❡r♠✐♥❛r ❛ s♦❧✉çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐

     

    

F1 = 1

F2 = 1

Fn =Fn−1+Fn−2, s❡n≥3

❈♦♠♦ ❛ r❡❝♦rrê♥❝✐❛ é ❤♦♠♦❣ê♥❡❛✱ ❧✐♥❡❛r ❞❡ ♦r❞❡♠ ✷✱ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s ♣r✐♠❡✐✲ r❛♠❡♥t❡ ❡♥❝♦♥tr❛♠♦s ❛s r❛í③❡s ❞♦ ♣♦❧✐♥ô♠✐♦p(x) =x2

−c1x−c2✱ ♦♥❞❡ ♥❡st❡ ❝❛s♦c1 =c2 = 1✳

❈♦♠♦ x2

−x₋1 = 0✱ ❛s r❛í③❡s sã♦r1 =

1 +√5

2 ❡ r2 =

1₋√5

2 ✳

P♦rt❛♥t♦Fn=α1

1 +√5 2

!n

+α2

1₋√5 2

!n

✳

P♦❞❡♠♦s ❞❡✜♥✐r F0 = 0 ❡ ❛✐♥❞❛ t❡r❡♠♦s F2 =F1+F0✳

❙❡❣✉❡ ❞❛í q✉❡ F0 =α1+α2 = 0 ❡ F1 =α1

1 +√5 2 +α2

1₋√5 2 = 1✳

❘❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛ ❡♥❝♦♥tr❛♠♦sα1 =

1

√

5 ❡ α2 =− 1

√

5✳

P♦rt❛♥t♦✱

Fn=

1

√

5

1 +√5 2

!n

− √1

5

1−√5 2

!n

✷✻

❈❛♣ít✉❧♦ ✸

❚r✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧

◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦ ❚r✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✱ ❢♦r♠❛❞♦ ♣❡❧♦s ❈♦❡✜❝✐❡♥t❡s ❇✐♥♦✲ ♠✐❛✐s✳ ❖ ❚r✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧ ❢♦✐ ❞❡✜♥✐❞♦ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❝❤✐♥ês ❨❛♥❣ ❍✉✐ ❡ ✈ár✐❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❢♦r❛♠ ❡st❛❜❡❧❡❝✐❞❛s ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ❇❧❛✐s❡ P❛s❝❛❧✳

✸✳✶ ❈♦❡✜❝✐❡♥t❡s ❇✐♥♦♠✐❛✐s

▲❡♠❛ ✶ ✭❘❡❧❛çã♦ ❞❡ ❙t✐❢❡❧✮

P❛r❛ t♦❞♦s n, i∈N✱ ❝♦♠ _i≤n ✈❛❧❡ q✉❡



 

n

i₋1 

 +



 

n

i



 =



 

n+ 1

i



 

Pr♦✈❛ ✶

✷✼

❚❡♠♦s q✉❡



 

n

i₋1 

 +



 

n

i



 =

n!

(i₋1)!(n₋i+ 1)! +

n!

i!(n₋i)! =

i.n! +n!(n−i+ 1)

i!(n₋i+ 1)! =

= n!(n+ 1)

i!(n−i+ 1)! =

(n+ 1)!

i!(n−i+ 1)! = 

 

n+ 1

i



 

Pr♦✈❛ ✷

❋❛r❡♠♦s ♦✉tr❛ ♣r♦✈❛ ✉s❛♥❞♦ ♦ s✐❣♥✐✜❝❛❞♦ ❞♦s sí♠❜♦❧♦s ✳

❙❡❥❛ A ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ n+ 1 ❡❧❡♠❡♥t♦s✳ ❈♦♥s✐❞❡r❡ a✱ ❛r❜✐trár✐♦✱ ✉♠ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ A✳

❚♦♠❛♥❞♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ q✉❛❧q✉❡r ♥❛ t❛❜❡❧❛ q✉❡ ❝♦♥té♠ t♦❞♦s ♦s



 

n+ 1

i





 s✉❜❝♦♥✲

❥✉♥t♦s ❝♦♥t❡♥❞♦ ❡①❛t❛♠❡♥t❡i❡❧❡♠❡♥t♦s ❡①✐st❡♠ ❛♣❡♥❛s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ✿ a ❡stá ♣r❡s❡♥t❡

♦✉a ♥ã♦ ❡stá ♣r❡s❡♥t❡✳

P♦rt❛♥t♦✱ s❡ s♦♠❛r♠♦s ♦ ♥ú♠❡r♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s q✉❡ ❝♦♥té♠ a✱ q✉❡ sã♦ ❡♠ ♥ú♠❡r♦



 

n

i₋1 



 ✱ ❝♦♠ ♦s q✉❡ ♥ã♦ ❝♦♥té♠ a✱ q✉❡ sã♦ ❡♠ ♥ú♠❡r♦ 

 

n

i





✱ t❡r❡♠♦s ♦ r❡s✉❧t❛❞♦ 

 

n+ 1

i



 ✳

✷✽

❚❡♦r❡♠❛ ✶ ✭❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥✮ P❛r❛ t♦❞♦s a, b_∈R❡ _n∈N ✈❛❧❡ q✉❡

(a+b)n ₌

   n 0   a n₊    n 1   a

n−1

b+    n 2   a

n−2

b2 +_{· · ·}+    n

n₋1 

 ab

n−1

+    n n   b n Pr♦✈❛

❆ ♣r♦✈❛ s❡rá ❢❡✐t❛ ♣❡❧♦ P■▼✳

P❛ss♦ ✶ P❛r❛ n= 1✱ t❡♠♦s(a+b)1

=a+b ❡

   1 0   a 1 +    1 1   a

1−1

b=a+b✱ ♣♦rt❛♥t♦

❛ ❢ór♠✉❧❛ é ✈á❧✐❞❛✳

P❛ss♦ ✷ ❙✉♣♦♥❤❛♠♦s ❛ ❢ór♠✉❧❛ ✈á❧✐❞❛ ♣❛r❛ n=k _≥1✱ ✐st♦ é✱

(a+b)k₌

   k 0   ak+

   k 1   ak

−1 b+    k 2   ak

−2 b2 +_{· · ·}+    k

k−1 

 abk

−1 +    k k   bk

❙❡❣✉❡ q✉❡(a+b)k+1

= (a+b)k_{.(a+b) = (a+b)}k_{.a+ (a+b)}k_.b=

=    k 0   ak

+1 + +    k 1   akb+

   k 2   ak

−1 b2 +    k 3   ak

−2 b3 +_{· · ·}+    k

k−1 

 a

2

bk−1

+    k k   abk+

+    k 0   a k b+    k 1   a

k−1

b2 +    k 2   a

k−2

b3 +· · ·+    k

k₋2 

 a

2

bk−1

+ 

 

k

k₋1 

 ab

k

✷✾

❯s❛♥❞♦ ❛❣♦r❛ ❛ r❡❧❛çã♦ ❞❡ ❙t✐❢❡❧ ♦❜t❡♠♦s

(a+b)k+1

= 

 

k+ 1

0   a k+1 +   

k+ 1

1 

 a

k_b+



 

k+ 1

2 

 a

k−1

b2

+_{· · ·}+ 

 

k+ 1

k+ 1 

 b

k+1

P♦rt❛♥t♦✱ ❛ ❢ór♠✉❧❛ ✈❛❧❡ ♣❛r❛ k+ 1✳

▲✐st❛♠♦s ❛ s❡❣✉✐r✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✱ ❛ ❡①♣❛♥sã♦ ❞❡

(a+b)n ₌

   n 0   an+

   n 1   an

−1 b+    n 2   an

−2 b2 +_{· · ·}+    n

n−1 

 abn

−1 +    n n   bn

(a+b)0

= 1

(a+b)1

=a+b

(a+b)2

=a2

+ 2ab+b2

(a+b)3

=a3

+ 3a2

b+ 3ab2

+b3

(a+b)4

=a4

+ 4a3

b+ 6a2

b2

+ 4ab3

+b4

(a+b)5

=a5

+ 5a4

b+ 10a3

b2

+ 10a2

b3

+ 5ab4

+b5

(a+b)6

=a6

+ 6a5

b+ 15a4

b2

+ 20a3

b3

+ 15a2

b4

+ 6ab5

✸✵

✸✳✷ ❚r✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧

▼♦♥t❛♥❞♦ ✉♠❛ t❛❜❡❧❛ ❝♦♠ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡st❛s ❡①♣❛♥sõ❡s ♦❜t❡♠♦s ♦ ❝❤❛♠❛❞♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✱ ♦♥❞❡ ❛ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛ é ♣r❡❡♥❝❤✐❞❛ s❡♠♣r❡ ❝♦♠ ♦ ♥ú♠❡r♦ ✶✱ ♦ ú❧t✐♠♦ ♥ú♠❡r♦ ❞❡ ❝❛❞❛ ❧✐♥❤❛ t❛♠❜é♠ é s❡♠♣r❡ ✶ ❡ ♦❜s❡r✈❛♥❞♦ q✉❡ s♦♠❛♥❞♦ ❞♦✐s ❡❧❡♠❡♥t♦s ❝♦♥s❡❝✉t✐✈♦s ❞❡ ✉♠❛ ❧✐♥❤❛ ♦❜t❡♠♦s ♦ ❡❧❡♠❡♥t♦ s✐t✉❛❞♦ ♥❛ ♠❡s♠❛ ❝♦❧✉♥❛ ❞♦ s❡❣✉♥❞♦ ❡ ♥✉♠❛ ❧✐♥❤❛ ❛❜❛✐①♦ ✭r❡❧❛ç❛♦ ❞❡ ❙t✐❢❡❧✮✿

✵ ✶ ✷ ✸ ✹ ✺ ✻ · · · n

✸✶

❙✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❞♦s ❝♦❡✜❝✐❡♥t❡s✿

✵ ✶ ✷ ✸ ✹ ✺ ✻

✵ ✶ ✶ ✶ ✶ ✷ ✶ ✷ ✶

✸ ✶ ✸ ✸ ✶

✹ ✶ ✹ ✻ ✹ ✶

✺ ✶ ✺ ✶✵ ✶✵ ✺ ✶ ✻ ✶ ✻ ✶✺ ✷✵ ✶✺ ✻ ✶

❉❡st❡ ♠♦❞♦ ♣♦❞❡♠♦s ❝♦♥t✐♥✉❛r ♦ ♣r❡❡♥❝❤✐♠❡♥t♦ ❞❛ t❛❜❡❧❛✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❙t✐❢❡❧✿

✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵

✵ ✶ ✶ ✶ ✶

✷ ✶ ✷ ✶

✸ ✶ ✸ ✸ ✶

✹ ✶ ✹ ✻ ✹ ✶

✺ ✶ ✺ ✶✵ ✶✵ ✺ ✶

✻ ✶ ✻ ✶✺ ✷✵ ✶✺ ✻ ✶

✼ ✶ ✼ ✷✶ ✸✺ ✸✺ ✷✶ ✼ ✶

✽ ✶ ✽ ✷✽ ✺✻ ✼✵ ✺✻ ✷✽ ✽ ✶

✾ ✶ ✾ ✸✻ ✽✹ ✶✷✻ ✶✷✻ ✽✹ ✸✻ ✾ ✶

✸✷

❊①❡♠♣❧♦ ✶✻ ❆ ú❧t✐♠❛ ❧✐♥❤❛ ❞❛ t❛❜❡❧❛ ❛❝✐♠❛✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦s ❞á ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡

(a+b)10

=a10

+10a9

b+45a8

b2

+120a7

b3

+210a6

b4

+252a5

b5

+210a4

b6

+120a3

b7

+45a2

b8

+10ab9

+b10

❙♦♠❛♥❞♦ ♦s t❡r♠♦s ❞❡ ✉♠❛ ❧✐♥❤❛ q✉❛❧q✉❡r ❞❡st❡ tr✐â♥❣✉❧♦✱ ♦❜s❡r✈❛♠♦s q✉❡✿



 

n

0 

 +



 

n

1 

 +



 

n

2 



+· · ·+ 

 

n

n−1 

 +



 

n

n



✸✸

❈❛♣ít✉❧♦ ✹

◆ú♠❡r♦s ❞❡ ❙t✐r❧✐♥❣

◆❡st❡ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐r❡♠♦s ♦s ♥ú♠❡r♦s ❞❡ ❙t✐r❧✐♥❣ ❞❡ Pr✐♠❡✐r❛ ❡ ❙❡❣✉♥❞❛ ❊s♣é❝✐❡ ❡ ❛♣r❡✲ s❡♥t❛r❡♠♦s s✉❛s r❡❧❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛✳ ❋❛r❡♠♦s t❛♠❜é♠✱ ❝♦♠♦ ♥♦ ❚r✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✱ ❛ ❝♦♥str✉çã♦ ❞❡ t❛❜❡❧❛s ❡♠ ❢♦r♠❛ ❞❡ tr✐â♥❣✉❧♦s ❡✱ ✉s❛♥❞♦ ❛s r❡❧❛çõ❡s ❞❡ r❡❝♦rrê♥❝✐❛✱ ✈❡r❡♠♦s ❝♦♠♦ ❛❝r❡s❝❡♥t❛r ❧✐♥❤❛s ♥❛ t❛❜❡❧❛✳

❊①❡♠♣❧♦ ✶✼ ❈♦♠❡ç❛r❡♠♦s ❛♥❛❧✐s❛♥❞♦ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿

❯♠❛ ♣r♦❢❡ss♦r❛ ♣❡❞❡ ♣❛r❛ ✷ ❞❡ s❡✉s ❛❧✉♥♦s q✉❡ ❛ ❛❥✉❞❡♠ ❛ ❝♦❧♦r✐r ❛s ❜❛♥❞❡✐r❛s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ♣❛r❛ ❛ ❞❡❝♦r❛çã♦ ❞❡ ✉♠❛ ❢❡st❛✳ ❆ ♣r♦❢❡ss♦r❛ ♣♦ss✉✐ ✹ ❝♦r❡s ❞❡ t✐♥t❛ ♣❛r❛ ❞✐str✐❜✉✐r ❛♦s s❡✉s ✷ ❛❧✉♥♦s✿ ❱❡r❞❡✱ ❆③✉❧✱ ▲❛r❛♥❥❛ ❡ Pr❡t♦✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❛ ♣r♦❢❡ss♦r❛ ♣♦❞❡rá ❞✐str✐❜✉✐r ❡ss❛s ❝♦r❡s ❞❡ t✐♥t❛ ♣❛r❛ s❡✉s ❛❧✉♥♦s ✭♥ã♦ s❡ ❢❛rá ❞✐st✐♥çã♦ ❡♥tr❡ ♦s ❛❧✉♥♦s✱ só ✐♠♣♦rt❛♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ ❞❛s ❝♦r❡s ❡♥tr❡ ❡❧❡s✮✱ ❞❡ ♠♦❞♦ q✉❡ ❝❛❞❛ ✉♠ ♣♦ss✉❛ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❝♦r ❞❡ t✐♥t❛ ❝♦♠ ❛ q✉❛❧ ♣♦ss❛ ❝♦❧♦r✐r ❛s ❜❛♥❞❡✐r❛s ❄