❊st✉❞♦ ❞♦ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥
♣♦r
❙❛❧❛t✐❡❧ ❉✐❛s ❞❛ ❙✐❧✈❛
❊st✉❞♦ ❞♦ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥
♣♦r
❙❛❧❛t✐❡❧ ❉✐❛s ❞❛ ❙✐❧✈❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞❛
Pr♦❢
❛✳ ❉r
❛✳ ❊❧✐s❛♥❞r❛ ❞❡ ❋át✐♠❛ ●❧♦ss ❞❡ ▼♦r❛❡s
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚✴❉▼✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❆❣♦st♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇
† ❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
❆❣r❛❞❡❝✐♠❡♥t♦s
◗✉❡r♦ ❛❣r❛❞❡❝❡r ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣❡❧♦ ❞♦♠ ❞❛ ✈✐❞❛ ❡ ❞❛ s❛❜❡❞♦r✐❛✳ ❆♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s q✉❡ s❡♠♣r❡ t♦r❝❡r❛♠ ❡♠ ♣r♦❧ ❞♦ ♠❡✉ ❝r❡s❝✐♠❡♥t♦ ❛❝❛❞ê♠✐❝♦✳ ❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s✱ q✉❡ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♣r♦❝❡ss♦ s❡ ♠♦str❛r❛♠ ❞✐s♣♦♥í✈❡✐s ❛ ❝♦♠♣❛rt✐❧❤❛r ❞❡ s❡✉s s❛❜❡r❡s✳ ➚ Pr♦❢❛✳ ❉r❛✳ ❊❧✐s❛♥❞r❛ ●❧♦ss ♣❡❧❛ s✉❛ ♦r✐❡♥t❛çã♦
❉❡❞✐❝❛tór✐❛
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ ✈❡♠ ♠♦str❛r ♦ ❡st✉❞♦ ❞♦s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ❜✐♥♦♠✐❛✐s ✐♥✐❝✐❛❞♦ ♥❛
7a sér✐❡ ✭8o ❛♥♦✮ ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✱ q✉❛♥❞♦ tr❛t❛♠♦s ❞❡ ♣r♦❞✉t♦s ♥♦tá✈❡✐s✱
q✉❡ é ❝♦♠♣❧❡♠❡♥t❛❞♦ ♥❛ s❡❣✉♥❞❛ sér✐❡ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❛ ♣❛rt✐r ❞♦ ❡st✉❞♦ ❞♦ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥✳ ❋❛r❡♠♦s ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞♦ ♠❡s♠♦✱ ♣❛ss❛♥❞♦ ♣♦r ✉♠ ❛♣❛♥❤❛❞♦ ❤✐stór✐❝♦ s♦❜r❡ ♦ ❛ss✉♥t♦✱ ♣r♦♣r✐❡❞❛❞❡s ❞♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦ ✭tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✴❚❛rt❛❣❧✐❛✮✱ ❝❤❡❣❛♥❞♦ ❛♦ ❚❡♦r❡♠❛ ❜✐♥♦♠✐❛❧ ❡✱ ♣♦r ✜♠✱ ❛ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡st❡s ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞✐✈❡rs♦s✱ ❡①♣❛♥sã♦ ♠✉❧t✐♥♦♠✐❛❧ ❡ ♥❛s sér✐❡s ❜✐♥♦♠✐❛✐s✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥✱ ❚r✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦✱ ❆♣❧✐❝❛çõ❡s✱ ❙ér✐❡s ❇✐♥♦♠✐❛✐s✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ❞❡❛❧s ✇✐t❤ t❤❡ st✉❞② ♦❢ t❤❡ ❜✐♥♦♠✐❛❧ ❞❡✈❡❧♦♣♠❡♥ts st❛rt❡❞ ✐♥ t❤❡ ❧❛t❡ ②❡❛rs ♦❢ ❊❧❡♠❡♥t❛r② ❙❝❤♦♦❧✱ ✇❤❡♥ ✇❡ ❞❡❛❧ ✇✐t❤ ♥♦t❛❜❧❡ ♣r♦❞✉❝ts✱ ✇❤✐❝❤ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡ s❡❝♦♥❞ ②❡❛r ♦❢ ❍✐❣❤ ❙❝❤♦♦❧✱ ❢r♦♠ t❤❡ st✉❞② ♦❢ ◆❡✇t♦♥✬s ❇✐♥♦♠✐❛❧✳ ❲❡ ✇✐❧❧ ♠❛❦❡ ❛ ❞❡t❛✐❧❡❞ st✉❞② ♦❢ t❤❡ s❛♠❡✱ t❤r♦✉❣❤ ❛ ❤✐st♦r✐❝❛❧ ♦✈❡r✈✐❡✇ ❛❜♦✉t t❤❡ s✉❜❥❡❝t✱ ♣r♦♣❡rt✐❡s ♦❢ ❛r✐t❤♠❡t✐❝ tr✐❛♥❣❧❡ ✭P❛s❝❛❧✬s tr✐❛♥❣❧❡ ✴ ❚❛rt❛❣❧✐❛✬s✮✱ r❡❛❝❤✐♥❣ t❤❡ ❜✐♥♦♠✐❛❧ t❤❡♦r❡♠ ❛♥❞✱ ✜♥❛❧❧②✱ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡s❡ r❡s✉❧ts ✐♥ s♦❧✈✐♥❣ ✈❛r✐♦✉s ♣r♦❜❧❡♠s✱ ✐♥ t❤❡ ♠✉❧t✐♥♦♠✐❛❧ ❡①♣❛♥❞✐♥❣ ❛♥❞ ✐♥ t❤❡ ❜✐♥♦♠✐❛❧ s❡r✐❡s✳
❑❡②✇♦r❞s✿ ◆❡✇t♦♥✬s ❇✐♥♦♠✐❛❧✱ ❆r✐t❤♠❡t✐❝ ❚r✐❛♥❣❧❡✱ ❆♣♣❧✐❝❛t✐♦♥s✱ ❇✐♥♦♠✐❛❧ ❙❡r✐❡s✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥ ✻
✶✳✶ ❘❡❧❡♠❜r❛♥❞♦ ♦ ❡st✉❞♦ ❞♦s Pr♦❞✉t♦s ◆♦tá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ◆ú♠❡r♦s ✭❝♦❡✜❝✐❡♥t❡s✮ ❜✐♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❖ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹ ❙♦♠❛tór✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✺ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡(a+b)n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✶✳✺✳✶ ❚❡♦r❡♠❛ ❞❡ ◆❡✇t♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✺✳✷ ❙♦♠❛ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✻ ❚❡r♠♦ ❣❡r❛❧ ❞♦ ❜✐♥ô♠✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✷ ❆♣❧✐❝❛çõ❡s ❞♦ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥ ✸✶
✷✳✶ ❊①♣❛♥sã♦ ▼✉❧t✐♥♦♠✐❛❧ ✲ P♦❧✐♥ô♠✐♦ ❞❡ ▲❡✐❜♥✐③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷ Pr♦❜❛❜✐❧✐❞❛❞❡s ✲ ▼ét♦❞♦ ❇✐♥♦♠✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✸ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥ ❛♣❧✐❝❛❞♦ à ●❡♥ét✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✹ ❙ér✐❡s ❇✐♥♦♠✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
❆ ❆❧❣✉♥s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ✺✹
❆✳✶ ❆①✐♦♠❛ ❞❛ ■♥❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
❆✳✷ ❉❡✜♥✐çã♦ ❞❡ ▲✐♠✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ❆✳✸ ❉❡✜♥✐çã♦ ❞❡ ❉❡r✐✈❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ❆✳✹ ❉❡r✐✈❛çã♦ ❞❡ ❙ér✐❡s ❞❡ P♦tê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✾
■♥tr♦❞✉çã♦
❆s r❡s♦❧✉çõ❡s ❞❡ ♣r♦❜❧❡♠❛s ❡ ♦ ✉s♦✱ ♥❡st❛s✱ ❞❡ ❢❡rr❛♠❡♥t❛s ❛❧❣é❜r✐❝❛s✱ s❡♠♣r❡ ❞❡s♣❡rt❛♠ ❛❧❣✉♠❛s ❞✐✜❝✉❧❞❛❞❡s ❡ ❞ú✈✐❞❛s ♥❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞♦s ❡❞✉❝❛♥❞♦s✳ ❙❡♥❞♦ ❛ss✐♠✱ é ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ q✉❡ ♦ ❛❧✉♥♦ ❛❞q✉✐r❛ ❤❛❜✐❧✐❞❛❞❡s ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ♣r♦❜❧❡♠❛s q✉❡ ♥❡❝❡ss✐t❛♠ ❞❡ ✉♠ tr❛t❛♠❡♥t♦ ❛❧❣é❜r✐❝♦ ♠❛✐s ❛♣✉r❛❞♦✳ ■st♦ tr❛rá ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ s♦❜r❡ ❞❡t❡r♠✐♥❛❞♦s ♣r♦❜❧❡♠❛s ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❞❡ ár❡❛s ❛✜♥s q✉❡ ✉s❛♠ ♦ tr❛t❛♠❡♥t♦ ❛❧❣é❜r✐❝♦ ❝♦♠♦ ❛❧✐❝❡r❝❡ ♣❛r❛ s✉❛s ❞❡♠♦♥str❛çõ❡s✳ ■♥✐❝✐❛r❡♠♦s✱ ♥♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❝♦♠ ♦ ❡st✉❞♦ ❞❡ ❡❧❡♠❡♥t♦s r❡❧❡✈❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧✱ ❢❛③❡♥❞♦ ✉s♦ ❞❡ ❢❡rr❛♠❡♥t❛s ♠❛t❡♠át✐❝❛s ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ♦♣❡r❛tór✐❛s q✉❡ sã♦ ♣❡rt✐♥❡♥t❡s ❛♦ ❡st✉❞♦ ❞♦ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥ ❡ ✐♥✈❡st✐❣❛♥❞♦ ❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❞❡st❛s ❡✱ ❛✐♥❞❛✱ é ✐♠♣♦rt❛♥t❡ s❛❧✐❡♥t❛r q✉❡✱ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ t❡ór✐❝♦✱ s❡rã♦ ♥❡❝❡ssár✐❛s ❛s ✐♥s❡rçõ❡s ❞❡ ❛❧❣✉♥s tó♣✐❝♦s q✉❡ tê♠ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛s ❞❡♠♦♥str❛çõ❡s q✉❡ s❡rã♦ ❢❡✐t❛s✱ ✉♠❛ ✈❡③ q✉❡ ♦ tr❛t♦ ❛❧❣é❜r✐❝♦ ✉t✐❧✐③❛❞♦ ♥❛s ❞❡♠♦♥str❛çõ❡s ❞❡s♣❡rt❛ ♥♦ ❧❡✐t♦r ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❝♦♠♣r❡❡♥sã♦ ❡ ❡①❡r❝✐t❛ ♦ ♣♦❞❡r ❞❡ ❛❜str❛çã♦ ❞♦ ♠❡s♠♦✳
❖ ❡st✉❞♦ ❞♦ ❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥ ❛❜r❡ ❝❛♠✐♥❤♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ✈ár✐♦s ♦✉tr♦s tó♣✐❝♦s ♠❛t❡♠át✐❝♦s✱ ❝♦♠♦ ♣♦❧✐♥ô♠✐♦s ❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✱ ❝♦♠♦ t❛♠❜é♠ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❡ ✐♥t❡❣r❛❧ ❝♦♠ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ❣r❛✉s ❞✐✈❡rs♦s✱ ♣♦✐s ❛tr❛✈és ❞❛s ❤❛❜✐❧✐❞❛❞❡s ❛❞q✉✐r✐❞❛s ❝♦♠ ♦ tr❛t♦ ❛❧❣é❜r✐❝♦✱ t♦r♥❛✲s❡ ♠❛✐s ❝♦♠♣r❡❡♥sí✈❡❧ ♦ ❞❡s❡♥✈♦❧✈❡r ❞❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❞❡♠♦♥str❛çõ❡s ❞❡st❛s✳ ❊st❡ ❡st✉❞♦✱ ❜❛s✐❝❛♠❡♥t❡ ❧✐t❡rár✐♦✱ ❢♦✐ ♣❡♥s❛❞♦ ❞❡ ❢♦r♠❛ ❛ ♦❢❡r❡❝❡r ❛♦ ❧❡✐t♦r
✉♠❛ s❡q✉ê♥❝✐❛ ❝❧❛r❛ s♦❜r❡ ❛ ❝♦♥str✉çã♦ ❞♦s s❛❜❡r❡s ♥❡❝❡ssár✐♦s ♣❛r❛ s❡ ❝❤❡❣❛r ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a+b)n ♣❛r❛ q✉❛❧q✉❡r ✈❛❧♦r ♥❛t✉r❛❧ ❞❡ n✳
❱❛❧❡ s❛❧✐❡♥t❛r t❛♠❜é♠ q✉❡✱ ❛♦ ❧♦♥❣♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❝♦♥t❡ú❞♦✱ s❡rã♦ ♣r♦♣♦st♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s✱ q✉❡ tê♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❞❛r ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❛♦s s❛❜❡r❡s ❛q✉✐ ❞❡s❡♥✈♦❧✈✐❞♦s✳
❊♠❜♦r❛ r❡❝❡❜❛ ♦ ♥♦♠❡ ❞❡ ✏❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥✑✱ ♦ ❡st✉❞♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧ ❝♦♠ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ❛♣❧✐❝❛çõ❡s✱ ♥ã♦ ❢♦✐ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❞♦ ❢ís✐❝♦ ❡ ♠❛t❡♠át✐❝♦ ✐♥❣❧ês ■s❛❛❝ ◆❡✇t♦♥ ✭✶✻✹✷ ✲ ✶✼✷✼✮✳ ◆❛ ✈❡r❞❛❞❡✱ s❡❣✉♥❞♦ ❬✷❪✱ ■s❛❛❝ ◆❡✇t♦♥ ❢❡③ ❛ ❞❡s❝r✐çã♦ ❡ ❡①♣❧✐❝❛çã♦ ❞♦ t❡♦r❡♠❛ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❜✐♥ô♠✐♦ ❣❡♥❡r❛❧✐③❛❞♦✱ ♣❛r❛ ♣♦tê♥❝✐❛s ❝♦♠ ❡①♣♦❡♥t❡ ❢r❛❝✐♦♥ár✐♦✱ ♦ q✉❛❧ ♦ ♠❡s♠♦ r❡♣r❡s❡♥t♦✉ s♦❜ ❛ ❢♦r♠❛
(P +P Q)m/n =Pm/n+m
nAQ+
m−n
2n BQ+
m−2n
3n CQ+..., ♦♥❞❡ ❆ r❡♣r❡s❡♥t❛ ♦ ♣r✐♠❡✐r♦ t❡r♠♦ Pm/n✱ ❇ r❡♣r❡s❡♥t❛ ♦ s❡❣✉♥❞♦ t❡r♠♦m
nAQ
✱ ❈ r❡♣r❡s❡♥t❛ ♦ t❡r❝❡✐r♦ t❡r♠♦ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ❖s ❞❡✈✐❞♦s ❛❥✉st❡s✱ ❝♦♠ s✉❛s ❞❡✈✐❞❛s r❡str✐çõ❡s✱ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧ ♣❛r❛ q✉❛❧q✉❡r ✈❛❧♦r ❝♦♠♣❧❡①♦ ❞♦ ❡①♣♦❡♥t❡✱ só ❢♦✐ ❡st❛❜❡❧❡❝✐❞♦ ♠❛✐s ❞❡ ✉♠ sé❝✉❧♦ ❡ ♠❡✐♦ ❞❡♣♦✐s ♣❡❧♦ ♠❛t❡♠át✐❝♦ ♥♦r✉❡❣✉ês ◆✳ ❍✳ ❆❜❡❧ ✭✶✽✵✷ ✲ ✶✽✷✾✮✳
❖ ❡st✉❞♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧ ❡①✐❣❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s q✉❡ ❢♦r❛♠ s❡♥❞♦ ✐♥s❡r✐❞♦s ♣♦r ✈ár✐♦s ♠❛t❡♠át✐❝♦s ❛♦ ❧♦♥❣♦ ❞♦s ❛♥♦s✳ ◆ã♦ s❡ ♣♦❞❡ ❢❛❧❛r ❡♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧ s❡♠ t❡r ❝♦♥❤❡❝✐♠❡♥t♦ ♣ré✈✐♦ ❞❡ ❛♥á❧✐s❡ ❝♦♠❜✐♥❛tór✐❛✱ ✉♠❛ ✈❡③ q✉❡ ♦s ✏❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s✑ ✭❝♦❡✜❝✐❡♥t❡s ❞♦s t❡r♠♦s ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❜✐♥ô♠✐♦✮✱ sã♦ ♦❜t✐❞♦s ♣❡❧❛ ❢ór♠✉❧❛ q✉❡ ❢♦r♥❡❝❡ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♠❜✐♥❛çõ❡s ❞❡ n ♦❜❥❡t♦s t♦♠❛❞♦s p ❞❡ ❝❛❞❛ ✈❡③✳ ❊st❡s ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s ❢♦r❛♠ ❛ss✐♠ ❝❤❛♠❛❞♦s ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦ ▼✐❝❤❛❡❧ ❙t✐❢❡❧ ✭✶✹✽✻ ✲ ✶✺✻✼✮✱ ❡✱ ♣♦r s✉❛ ✈❡③✱ ❢♦r❛♠ ♦r❣❛♥✐③❛❞♦s ♥✉♠❛ t❛❜❡❧❛ ❡♠ ❢♦r♠❛ ❞❡ tr✐â♥❣✉❧♦✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦✱ tr✐â♥❣✉❧♦ ❞❡ ❨❛♥❣✲❍✉✐✱ ♦✉ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✴❚❛rt❛❣❧✐❛✳
❙❡❣✉♥❞♦ ❬✶❪✱ ♦ ❊s♣❡❧❤♦ Pr❡❝✐♦s♦ ❝♦♠❡ç❛ ❝♦♠ ✉♠ ❞✐❛❣r❛♠❛ ❞♦ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦✱ ♦♥❞❡ t❡♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ❡①♣❛♥sõ❡s ❜✐♥♦♠✐❛✐s ❛té ❛ ♦✐t❛✈❛ ♣♦tê♥❝✐❛✱ ❝❧❛r❛♠❡♥t❡ ❞❛❞❛s ❡♠ ♥✉♠❡r❛✐s ❡♠ ❜❛rr❛ ❡ ✉♠ sí♠❜♦❧♦ r❡❞♦♥❞♦ ♣❛r❛ ♦ ③❡r♦✳
❊♠ ♠❡❛❞♦s ❞♦ sé❝✉❧♦ ❳■ ✭♣♦r ✈♦❧t❛ ❞❡ ✶✵✺✵✮✱ ♦ ▼❛♥✉❛❧ ❞❡ ▼❛t❡♠át✐❝❛ ❞❡ ❏✐❛ ❳✐❛♥✱ ♥❛ ❈❤✐♥❛✱ ❥á tr❛③ ♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦✳ ❉❡♣♦✐s ❞✐ss♦✱ ♦ ♠❛t❡♠át✐❝♦ ❡ ❛strô♥♦♠♦ ♣❡rs❛ ❖♠❛r ❑❤❛②②❛♠ ✭✶✵✹✽ ✲ ✶✶✷✷✮✱ t❛♠❜é♠ ❢❡③ ♠❡♥çã♦ ❞♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦ ❡♠ ❛❧❣✉♥s ❞❡ s❡✉s tr❛❜❛❧❤♦s✱ ♣♦r ✈♦❧t❛ ❞❡ ✶✶✵✵✳
❋✐❣✉r❛ ✷✿ ❚r✐â♥❣✉❧♦ ❞❡ ❏✐❛ ❳✐❛♥
❯♠ sé❝✉❧♦ ❛♥t❡s ❞❡ P❛s❝❛❧✱ ♥❛ ❊✉r♦♣❛✱ ❛❧❣✉♥s ♠❛t❡♠át✐❝♦s t❛♠❜é♠ tr❛❜❛❧❤❛r❛♠ ❝♦♠ ♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦✳ ❉❡♥tr❡ ❡st❡s ♣♦❞❡♠♦s ❝✐t❛r ♦ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦ P❡tr✉s ❆♣✐❛♥✉s ✭✶✹✾✺ ✲ ✶✺✺✷✮✱ q✉❡ ♣✉❜❧✐❝♦✉ ✉♠ ❧✐✈r♦ q✉❡ tr❛③✐❛ ❡♠ s✉❛ ❝❛♣❛ ♦ ❞❡s❡♥❤♦ ❞♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦✱ ❡♠ ✶✺✷✼✳
❯♠ ❞♦s ♣r✐♠❡✐r♦s ♠❛t❡♠át✐❝♦s ♦❝✐❞❡♥t❛✐s ❛ ❝♦♥❢❡❝❝✐♦♥❛r ✉♠❛ t❛❜❡❧❛ ❝♦♥t❡♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♠❜✐♥❛çõ❡s ♣♦ssí✈❡✐s ♥✉♠ ❧❛♥ç❛♠❡♥t♦ ❞❡ ✉♠ ❞❛❞♦ ❢♦✐ ♦ ♠❛t❡♠át✐❝♦ ✐t❛❧✐❛♥♦ ◆✐❝❝♦❧♦ ❋♦♥t❛♥❛ ❚❛rt❛❣❧✐❛ ✭✶✹✾✾ ✲ ✶✺✺✾✮✱ s❡♥❞♦ ❛ss✐♠✱ ❚❛rt❛❣❧✐❛ r❡✐✈✐♥❞✐❝♦✉ ❛ ❝r✐❛çã♦ ❞♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦ ♣❛r❛ ❡❧❡✱ ♦ q✉❡ ❡①♣❧✐❝❛ ♦ ❢❛t♦ ❞❡ q✉❡ ❡♠ ❛❧❣✉♥s ♣❛ís❡s✱ ❛té ♦s ❞✐❛s ❞❡ ❤♦❥❡✱ ♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ tr✐â♥❣✉❧♦ ❞❡ ❚❛rt❛❣❧✐❛✳ P♦ré♠✱ ❝♦♠♦ ♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ❇❧❛✐s❡ P❛s❝❛❧ ✭✶✻✷✸ ✲ ✶✻✻✷✮ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❞❡s❝♦❜r✐❞♦r ❝♦♥❤❡❝✐❞♦ ❞♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦ ♥♦ ♦❝✐❞❡♥t❡ ❡ ❞❡✈✐❞♦ às ❛♣❧✐❝❛çõ❡s q✉❡ ♦ ♠❡s♠♦ ❢❛③✐❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡st❡✱ ♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦ ♣❛ss♦✉ ❛ s❡r ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✳ ❊st❡ r❡❝♦♥❤❡❝✐♠❡♥t♦ t♦r♥♦✉✲s❡ ♠❛✐s ♥♦tór✐♦ q✉❛♥❞♦✱ ❡♠ ✶✼✸✾✱ ❆❜r❛❤❛♠ ❞❡ ▼♦✐✈r❡ ✭✶✻✻✼ ✲ ✶✼✺✹✮ ♣✉❜❧✐❝♦✉ ✉♠ tr❛❜❛❧❤♦ ❞❡ ♠✉✐t♦ ✐♠♣❛❝t♦ ♥❛ é♣♦❝❛✱ ✐♥t✐t✉❧❛❞♦ ✏tr✐❛♥❣✉❧✉♠ ❛r✐t❤♠❡t✐❝✉♠ ♣❛s❝❛❧✐✉♠✑ s♦❜r❡ ♦ tr✐â♥❣✉❧♦
❖ t❡♦r❡♠❛ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧ t❛♠❜é♠ ❢♦✐ ❡st✉❞❛❞♦ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ✈♦♥ ▲❡✐❜♥✐③ ✭✶✻✹✻✲✶✼✶✻✮✳ ▲❡✐❜♥✐③✱ s❡❣✉♥❞♦ ❬✷❪✱ ❢❡③ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ t❡♦r❡♠❛ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧ ♣❛r❛ ♦ t❡♦r❡♠❛ ♠✉❧t✐♥♦♠✐❛❧✱ ♦ q✉❡ ❝♦♥s✐st❡ ❡♠ ❢❛③❡r ❛ ❡①♣❛♥sã♦ ❞❡ (a1+a2+...+an)r✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♠✉❧t✐♥♦♠✐❛❧ ✭P♦❧✐♥ô♠✐♦ ❞❡ ▲❡✐❜♥✐③✮✳ ❋❛r❡♠♦s ✉s♦ ❞♦ ♠ét♦❞♦ ❜✐♥♦♠✐❛❧ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s✱ ❛❧é♠ ❞❡ ❡st✉❞❛r ❛ ❛♣❧✐❝❛çã♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧ ♥♦ ❡st✉❞♦ ❞❡ ●❡♥ét✐❝❛ ❡✱ ♣♦r ✜♠✱ ♦ ✉s♦ ❞♦ ❚❡♦r❡♠❛ ❇✐♥♦♠✐❛❧ ❣❡♥❡r❛❧✐③❛❞♦ ♥♦ ❡st✉❞♦ ❞❛s sér✐❡s ❜✐♥♦♠✐❛✐s ❡ ♥❛ ❡①♣❛♥sã♦ ❞❡ ❢✉♥çõ❡s ❝♦♠♦ sér✐❡s ❞❡ ♣♦tê♥❝✐❛s✳
❊s♣❡r❛✲s❡✱ ♣♦rt❛♥t♦✱ q✉❡ ❡st❡ ❡st✉❞♦ s❡❥❛ ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛✱ ✉♠❛ ✈❡③ q✉❡ ♦ ♠❡s♠♦ s❡rá ❢✉♥❞❛♠❡♥t❛❞♦ ♥❛s r❡✢❡①õ❡s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❝❛❞❛ ✈❡③ ♠❛✐s s✐❣♥✐✜❝❛t✐✈♦✳
❈❛♣ít✉❧♦ ✶
❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❇✐♥ô♠✐♦ ❞❡
◆❡✇t♦♥
◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s q✉❡ sã♦ ❡ss❡♥❝✐❛✐s ♣❛r❛ ❢❛③❡r ❛ ❡①♣❛♥sã♦ ❜✐♥♦♠✐❛❧✱ ❛❧❣✉♥s ❞♦s q✉❛✐s s❡rã♦ ❞❡♠♦♥str❛❞♦s ❞❡ ♠♦❞♦ ❛ ♥♦s ♣r♦♣♦r❝✐♦♥❛r ✉♠ s✉♣♦rt❡ r❡❛❧ ♣❛r❛ ❛ ❛♣❧✐❝❛çã♦ ❞♦s ♠❡s♠♦s✳
✶✳✶ ❘❡❧❡♠❜r❛♥❞♦ ♦ ❡st✉❞♦ ❞♦s Pr♦❞✉t♦s ◆♦tá✈❡✐s
❈♦♠♦ ❥á ❢♦✐ ❞✐t♦✱ ♦ ❡st✉❞♦ ❞♦s ♣r♦❞✉t♦s ♥♦tá✈❡✐s é ❢❡✐t♦ ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ♠❡s♠♦ s❡ ❜❛s❡✐❛ ♥✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✏r❡❣r❛s✑ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a+b)n✱ ❝♦♠ n≤3✳
❙❡♥❞♦n ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✱ t❡♠♦s q✉❡✿
• n = 0 ⇒ (a+b)0= 1
• n = 1 ⇒ (a+b)1=a+b
• n = 2 ⇒ (a+b)2 = (a+b).(a+b) = a2+ 2ab+b2
• n = 3 ⇒ (a+b)3 = (a+b).(a+b)2 =a3+ 3a2b+ 3ab2+b3.
P❛r❛ ✈❛❧♦r❡s ❞❡ n ♠❛✐♦r❡s q✉❡ ✸✱ t❛♠❜é♠ ♣♦❞❡♠♦s ❢❛③❡r ✉s♦ ❞❛s r❡❣r❛s ❛❝✐♠❛ ❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♣♦tê♥❝✐❛s ♣❛r❛ ❝❤❡❣❛r ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❜✐♥ô♠✐♦✳ P♦r ❡①❡♠♣❧♦✿
(a+b)4 = (a+b).(a+b)3 = (a+b).(a3+ 3a2b+ 3ab2+b3)
= a4+ 4a3b+ 6a2b2+ 4ab3+b4.
❉❡ ♠♦❞♦ ❣❡r❛❧✱ (a+b)n ❂ (a+b)✳(a+b)n−1✳
❊ss❡ ♣r♦❝❡ss♦✱ ♣♦ré♠✱ s❡ t♦r♥❛ ❞✐❢í❝✐❧ ❡ ❞❡♠♦r❛❞♦✱ ♣♦✐s ♥♦s r❡♠❡t❡ ❛ ❝á❧❝✉❧♦s ♠✉✐t♦ tr❛❜❛❧❤♦s♦s✳ ❙❡♥❞♦ ❛ss✐♠✱ ❞✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✱ ♠♦str❛r❡♠♦s ❛❧❣✉♠❛s ❢❡rr❛♠❡♥t❛s ♠❛t❡♠át✐❝❛s q✉❡ t♦r♥❛rã♦ ♠❛✐s ♣rát✐❝♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a+b)n✳
✶✳✷ ◆ú♠❡r♦s ✭❝♦❡✜❝✐❡♥t❡s✮ ❜✐♥♦♠✐❛✐s
◆♦s t❡♠❛s ❛❜♦r❞❛❞♦s ❛ s❡❣✉✐r✱ ✉s❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ❢❛t♦r✐❛❧ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❡✱ ❛✐♥❞❛✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s N =
{0,1,2,3,4,5, ...}
❉❡✜♥❛♠♦s ❢❛t♦r✐❛❧ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ s❡❣✉♥❞♦ ❬✻❪✿
❉❡✜♥✐çã♦ ✶ ✿ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧n, n≥2✱ ♦ ❢❛t♦r✐❛❧ ❞❡ ♥ ✭✐♥❞✐❝❛✲s❡ ♣♦r n!✮
é ♦ ♣r♦❞✉t♦ ❞♦s ♥ ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣♦s✐t✐✈♦s✱ ❡s❝r✐t♦s ❞❡s❞❡ ♥ ❛té ✶✱ ✐st♦ é✿
n! =n·(n−1)·(n−2)... 3·2·1
P♦r ❡①❡♠♣❧♦✿
5! = 5·4·3·2·1 = 120
8! = 8·7·6·5·4·3·2·1 = 40320✳
◆ú♠❡r♦s ✭❝♦❡✜❝✐❡♥t❡s✮ ❜✐♥♦♠✐❛✐s ❈❛♣ít✉❧♦ ✶
❱❛❧❡♠ t❛♠❜é♠ ❛s s❡❣✉✐♥t❡s ❝♦♥✈❡♥çõ❡s ❡s♣❡❝✐❛✐s✿
0! = 1
e
1! = 1.
❊♠ ❛♥á❧✐s❡ ❝♦♠❜✐♥❛tór✐❛✱ ♦ ♥ú♠❡r♦Cn,p r❡♣r❡s❡♥t❛ ♦ t♦t❛❧ ❞❡ ❝♦♠❜✐♥❛çõ❡s ❞❡p
❡❧❡♠❡♥t♦s q✉❡ ♣♦❞❡♠♦s ❢♦r♠❛r ❛ ♣❛rt✐r ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ n ❡❧❡♠❡♥t♦s✳ ❙❡❣✉♥❞♦ ❬✶✵❪✱ ❝❛❞❛ s✉❜❝♦♥❥✉♥t♦ ❝♦♠ p ❡❧❡♠❡♥t♦s é ❝❤❛♠❛❞♦ ❞❡ ✉♠❛ ❝♦♠❜✐♥❛çã♦ s✐♠♣❧❡s ❞❡ ❝❧❛ss❡ p ❞♦s n ♦❜❥❡t♦s a1, a2, a3, . . . , an.
❱❡❥❛♠♦s✱ ❛❣♦r❛✱ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s✳
❉❡✜♥✐çã♦ ✷ ❈♦♥s✐❞❡r❡♠♦s ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s n ❡ p✱ s❡♥❞♦ n≥p✳ ❈❤❛♠❛♠♦s ❞❡ ❝♦❡✜❝✐❡♥t❡ ❜✐♥♦♠✐❛❧ n s♦❜r❡ p✱ ❡ ✐♥❞✐❝❛♠♦s ♣♦r
n p
✱ ♦ ♥ú♠❡r♦ ❞❛❞♦ ♣♦r
n p
= n!
p!(n−p)! = Cn,p.
❖ ♥ú♠❡r♦ n é ❝❤❛♠❛❞♦ ❞❡ ♥✉♠❡r❛❞♦r ❡ ♦ ♥ú♠❡r♦ p ❞❡ ❞❡♥♦♠✐♥❛❞♦r ❞❡
n p
✳
❙❡♥❞♦ ❛ss✐♠✱ t❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✿
•
6 2
= C6,2 =
6!
2!·(6−2)! = 6! 2!·4! =
6·5·4! 2!·4! = 15.
•
10 3
= C10,3 =
10!
3!·(10−3)! = 10! 3!·7! =
10·9·8·7!
3·2·7! = 120.
❖❜s❡r✈❡♠♦s ❛❧❣✉♥s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✿
• ❙❡ p= 0✱ t❡♠♦s
n
0
= n!
0!n! = 1✱∀ n∈N✳
❉❡ss❡ ♠♦❞♦✱ t❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱
3 0
= 1 ❡
17 0
= 1✳
• ❙❡ p= 1✱ t❡♠♦s
n
1
= n! 1!(n−1)! =
n·(n−1)!
(n−1)! = n✱∀ n ∈N✳
❉❡ss❡ ♠♦❞♦✱ t❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱
8 1
= 8 ❡
25 1
= 25✳
• ❙❡ p=n✱ t❡♠♦s
n n
= n!
n!0! = 1✱ ∀ n∈N✳
❉❡ss❡ ♠♦❞♦✱ t❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱
5 5
= 1 ❡
32 32
= 1✳
❉❡✜♥❛♠♦s ❛❣♦r❛ ♦s ♥ú♠❡r♦s ❜✐♥♦♠✐❛✐s ❝♦♠♣❧❡♠❡♥t❛r❡s✱ ♣❛r❛ ❢❛③❡r ✉s♦ ❞♦s ♠❡s♠♦s ❡♠ ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s ❛ s❡❣✉✐r✳
❉❡✜♥✐çã♦ ✸ ❉♦✐s ♥ú♠❡r♦s ❜✐♥♦♠✐❛✐s
n p ❡ n q
sã♦ ❝♦♠♣❧❡♠❡♥t❛r❡s s❡p+q =n✳
❈♦♠ ❡st❛ ❞❡✜♥✐çã♦✱ ♣❡r❝❡❜❡♠♦s q✉❡ sã♦ ❜✐♥♦♠✐❛✐s ❝♦♠♣❧❡♠❡♥t❛r❡s✱ ♣♦r ❡①❡♠♣❧♦✿
6 2 ❡ 6 4 ; 15 5 ❡ 15 10 ; 31 3 ❡ 31 28 .
❚❡♠♦s ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❢✉♥❞❛♠❡♥t❛❧ q✉❛♥❞♦ s❡ tr❛t❛ ❞❡ ❜✐♥♦♠✐❛✐s ❝♦♠♣❧❡♠❡♥t❛r❡s✳
Pr♦♣r✐❡❞❛❞❡✿ ❉♦✐s ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s ❝♦♠♣❧❡♠❡♥t❛r❡s tê♠ ♦ ♠❡s♠♦ ✈❛❧♦r✳
p+q=n ⇒
n p = n q .
❏✉st✐✜❝❛t✐✈❛✿ ❙❡♥❞♦✱ p+q =n✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡rp=n−q✳ ❆ss✐♠✱ s❡❣✉❡ q✉❡✿
n p = n n−q
= n!
(n−q)![n−(n−q)]! =
n! (n−q)!q! =
n q , ❝♦♠♦ ❞❡s❡❥❛❞♦✳
✶✳✸ ❖ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦
❖ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦✱ ❝✉❥❛ ❤✐stór✐❛ ❥á ❢♦✐ r❡❧❛t❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✱ ❝♦♥s✐st❡ ❡♠ ✉♠❛ t❛❜❡❧❛✱ ♥❛ q✉❛❧ ❡stã♦ ❞✐s♣♦st♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s✱ ❞❡ ♠♦❞♦ q✉❡ ♦s
❖ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦ ❈❛♣ít✉❧♦ ✶
❝♦❡✜❝✐❡♥t❡s ❞❡ ♠❡s♠♦ ♥✉♠❡r❛❞♦r ❛❣r✉♣❛♠✲s❡ ♥✉♠❛ ♠❡s♠❛ ❧✐♥❤❛ ❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ♠❡s♠♦ ❞❡♥♦♠✐♥❛❞♦r ❛❣r✉♣❛♠✲s❡ ♥❛ ♠❡s♠❛ ❝♦❧✉♥❛✿
❧✐♥❤❛ 0 −→
0 0
❧✐♥❤❛ 1 −→
1 0 1 1
❧✐♥❤❛ 2 −→
2 0 2 1 2 2
❧✐♥❤❛ 3 −→
3 0 3 1 3 2 3 3
❧✐♥❤❛ 4 −→
4 0 4 1 4 2 4 3 4 4 ✳✳✳ ❧✐♥❤❛ n −→
n 0 n 1 n 2 n 3 n 4 · · · n n . P❡r❝❡❜❛♠♦s q✉❡ ♦ t❡r♠♦ ❧✐♥❤❛n s✐❣♥✐✜❝❛ ❛ ❧✐♥❤❛ ❞❡ ♥✉♠❡r❛❞♦r n✳
❈❛❧❝✉❧❛♥❞♦ ♦s r❡s♣❡❝t✐✈♦s ✈❛❧♦r❡s ❞♦s ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s✱ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ r❡♣r❡s❡♥t❛çã♦ ♣❛r❛ ♦ tr✐â♥❣✉❧♦✿
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1
✳✳✳
✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦
Pr♦♣r✐❡❞❛❞❡ ✶ ❊♠ ❝❛❞❛ ❧✐♥❤❛ ❞♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦✱ ♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ é s❡♠♣r❡ ✐❣✉❛❧ ❛ 1✳
❏✉st✐✜❝❛t✐✈❛✿ ❉❡ ❢❛t♦✱ ♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ❞❛n✲és✐♠❛ ❧✐♥❤❛ ❞♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦ é
n
0
= 1✱ ∀ n∈N✳
Pr♦♣r✐❡❞❛❞❡ ✷ ❊♠ ❝❛❞❛ ❧✐♥❤❛ ❞♦ tr✐â♥❣✉❧♦✱ ♦ ú❧t✐♠♦ ❡❧❡♠❡♥t♦ é s❡♠♣r❡ ✐❣✉❛❧ ❛ 1✳
❏✉st✐✜❝❛t✐✈❛✿ ❉❡ ❢❛t♦✱ ♦ ú❧t✐♠♦ ❡❧❡♠❡♥t♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❧✐♥❤❛s ❞♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦ é
n n
= 1✱ ∀ n∈N✳
Pr♦♣r✐❡❞❛❞❡ ✸✳ ✭ P♦r ❬✶✵❪✮ ❙♦♠❛♥❞♦ ❞♦✐s ❡❧❡♠❡♥t♦s ❝♦♥s❡❝✉t✐✈♦s ❞❡ ✉♠❛ ♠❡s♠❛ ❧✐♥❤❛ ♦❜t❡♠♦s ♦ ❡❧❡♠❡♥t♦ s✐t✉❛❞♦ ❛❜❛✐①♦ ❞❛ ú❧t✐♠❛ ♣❛r❝❡❧❛✳
❋✐❣✉r❛ ✶✳✶✿ ❘❡❧❛çã♦ ❞❡ ❙t✐❢❡❧
❊st❛ ♣r♦♣r✐❡❞❛❞❡ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ r❡❧❛çã♦ ❞❡ ❙t✐❢❡❧ ✭♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦ ❥á ❝✐t❛❞♦ ♥❛ ✐♥tr♦❞✉çã♦ ❞❡st❡ tr❛❜❛❧❤♦✮ ❡ ❛✜r♠❛ q✉❡✿
n p
=
n−1
p−1
+
n−1
p
, n ≥p ❡ n ≥2.
❖ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦ ❈❛♣ít✉❧♦ ✶
❏✉st✐✜❝❛t✐✈❛✿ ❯t✐❧✐③❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s✱ t❡♠♦s✿
n−1
p−1
+
n−1
p
= (n−1)!
(p−1)![(n−1)−(p−1)]! +
(n−1)!
p!(n−1−p)! = (n−1)!
(p−1)!(n−p)!+
(n−1)!
p!(n−p−1)! = (n−1)!
(p−1)!(n−p)·(n−p−1)! +
(n−1)!
p·(p−1)!(n−p−1)! = (n−1)!
(p−1)!(n−p−1)! ·
1
n−p +
1
p
= (n−1)!
(p−1)!(n−p−1)! ·
n
(n−p)p
= n!
p!(n−p)! =
n p , ❝♦♠♦ ❛✜r♠❛❞♦✳
Pr♦♣r✐❡❞❛❞❡ ✹✳ ◆✉♠❛ ❧✐♥❤❛✱ ❞♦✐s ❜✐♥♦♠✐❛✐s ❡q✉✐❞✐st❛♥t❡s ❞♦s ❡①tr❡♠♦s tê♠ ♦ ♠❡s♠♦ ✈❛❧♦r✱ ♦✉ s❡❥❛✱
n p = n n−p
✳
❏✉st✐✜❝❛t✐✈❛✿ ❉❡ ❢❛t♦✱ ❞♦✐s ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s ❡q✉✐❞✐st❛♥t❡s ❞♦s ❡①tr❡♠♦s sã♦ ❝♦♠♣❧❡♠❡♥t❛r❡s ❡✱ ♣♦rt❛♥t♦✱ tê♠ ♦ ♠❡s♠♦ ✈❛❧♦r✳ ❙❡♥❞♦ ❛ss✐♠✱
n p
= n!
p!(n−p)! =
n!
(n−p)!(n−(n−p))! =
n n−p
,
❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳
Pr♦♣r✐❡❞❛❞❡ ✺ ✭❚❡♦r❡♠❛ ❞❛s ❧✐♥❤❛s✮✳ ❆ s♦♠❛ ❞♦s ❡❧❡♠❡♥t♦s ❞❛ n✲és✐♠❛ ❧✐♥❤❛ é s❡♠♣r❡ ✐❣✉❛❧ ❛ 2n✳
n 0 + n 1 + n 2 + n 3 + n 4 +· · ·+ n n
= 2n. ❏✉st✐✜❝❛t✐✈❛✿ ✭P♦r ✐♥❞✉çã♦✮
❙❡❣✉♥❞♦ ❬✽❪✱ s❡❥❛ P(n)❛ ♣r♦♣♦s✐çã♦✿ s❡ n≥0✱ ❡♥tã♦
Sn=
n 0 + n 1 + n 2 + n 3 + n 4 +· · ·+ n n
• n = 0⇒S0 =
0 0
= 1 = 20 X
• n = 1⇒S1 =
1 0 + 1 1
= 2 = 21 X
❙✉♣♦♠♦s ❛❣♦r❛ q✉❡ ❛ ♣r♦♣♦s✐çã♦ P(n)é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n= 0,1,2,3, . . . , k✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ P(n) ❝♦♥t✐♥✉❛ ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n =k+ 1, ✐st♦ é✱ Sk+1 =
k+ 1 0
+
k+ 1 1
+
k+ 1 2
+
k+ 1 3
+
k+ 1 4
+· · ·+
k+ 1
k+ 1
= 2k+1. ❚❡♠♦s q✉❡✱
Sk+1 =
k+ 1 0
+
k+ 1 1
+
k+ 1 2
+
k+ 1 3
+· · ·+
k+ 1
k
+
k+ 1
k+ 1
. ❆♣❧✐❝❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❙t✐❢❡❧✱ ❞❛ s❡❣✉♥❞❛ ♣❛r❝❡❧❛ ❞❛ s♦♠❛ ❛té ❛ ♣❡♥ú❧t✐♠❛ ❡✱ s❛❜❡♥❞♦ q✉❡
k+ 1 0 = k 0 ❡
k+ 1
k+ 1
= k k , t❡r❡♠♦s
Sk+1= k 0 + k 0 + k 1
| {z }
+ k 1 + k 2
| {z }
+ k 2 + k 3
| {z }
+· · ·+
k
k−1
+ k
k
| {z }
+ k k . ❖✉ s❡❥❛✱
Sk+1 =
k 0 + k 1 + k 2 +· · ·+ k k
| {z }
+ k 0 + k 1 + k 2 +· · ·+ k k
| {z }
= 2·Sk.
❈♦♠♦✱ ♣♦r ❤✐♣ót❡s❡✱ Sk = 2k✱ ❡♥tã♦
Sk+1 = 2·2k = 2k+1.
▲♦❣♦✱ P(k+ 1) é ✈❡r❞❛❞❡✐r❛ ❡ ❝♦♥❝❧✉í♠♦s✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ✶✵✱ q✉❡P(n) é ✈❡r❞❛❞❡✐r❛
♣❛r❛ t♦❞♦ n ♥❛t✉r❛❧✱ ❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳ ❖✉tr♦ ♠♦❞♦ ❞❡ ♣r♦✈❛r ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ é ♣❡❧♦ t❡♦r❡♠❛ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧✱ q✉❡ ✈❡r❡♠♦s ❡♠ ❜r❡✈❡✳
❖ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦ ❈❛♣ít✉❧♦ ✶
Pr♦♣r✐❡❞❛❞❡ ✻ ✭❚❡♦r❡♠❛ ❞❛s ❝♦❧✉♥❛s✮✳ ❆ s♦♠❛ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ❝♦❧✉♥❛ ❞♦ tr✐â♥❣✉❧♦✱ ❝♦♠❡ç❛♥❞♦ ❞♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ❞❛ ❝♦❧✉♥❛✱ t❡♠ ♦ ♠❡s♠♦ ✈❛❧♦r q✉❡ ♦ ❡❧❡♠❡♥t♦ q✉❡ s❡ ❡♥❝♦♥tr❛ ♥❛ ❧✐♥❤❛ ❡ ❝♦❧✉♥❛ ✐♠❡❞✐❛t❛♠❡♥t❡ ♣♦st❡r✐♦r ❛♦ ú❧t✐♠♦ ❝♦❡✜❝✐❡♥t❡ ❜✐♥♦♠✐❛❧ ❞❛ s♦♠❛✳
❋✐❣✉r❛ ✶✳✷✿ ❚❡♦r❡♠❛ ❞❛s ❝♦❧✉♥❛s
❚❡♠♦s✱ ♥❛ ❢♦r♠❛ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s✿
p p +
p+ 1
p
+
p+ 2
p
+
p+ 3
p
+· · ·+
p+n p
=
p+n+ 1
p+ 1
. ❏✉st✐✜❝❛t✐✈❛✿ ❱❛♠♦s ❛♣❧✐❝❛r ❛ r❡❧❛çã♦ ❞❡ ❙t✐❢❡❧ ❛♦s ❡❧❡♠❡♥t♦s ❞❛ ❝♦❧✉♥❛ p+ 1✱ ❛
♣❛rt✐r ❞❛ ❧✐♥❤❛ p+ 2✳ ▲❡♠❜r❛♠♦s q✉❡ pp = 1 ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ p✱ ♦ q✉❡
♥♦s ❢♦r♥❡❝❡ ❛ ♣r✐♠❡✐r❛ ❞❛s ✐❣✉❛❧❞❛❞❡s ❛ s❡❣✉✐r✳
p+ 1
p+ 1
= p p
p+ 2
p+ 1
=
p+ 1
p+ 1
+
p+ 1
p
p+ 3
p+ 1
=
p+ 2
p+ 1
+
p+ 2
p
✳✳✳
p+n p+ 1
=
p+n−1
p+ 1
+
p+n−1
p
p+n+ 1
p+ 1
=
p+n p+ 1
+
p+n p
❋❛③❡♥❞♦ ❛ s♦♠❛ ❞❡ t♦❞❛s ❛s r❡❧❛çõ❡s ❡s❝r✐t❛s ❡ s✐♠♣❧✐✜❝❛♥❞♦ ❛s ♣❛r❝❡❧❛s ✐❣✉❛✐s q✉❡ ❛♣❛r❡❝❡♠ ❡♠ ♠❡♠❜r♦s ♦♣♦st♦s✱ t❡r❡♠♦s
p+n+ 1
p+ 1
= p p +
p+ 1
p
+
p+ 2
p
+
p+ 3
p
+· · ·+
p+n p
,
❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳
Pr♦♣r✐❡❞❛❞❡ ✼ ✭❚❡♦r❡♠❛ ❞❛s ❞✐❛❣♦♥❛✐s✮✳ ❆ s♦♠❛ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ❞✐❛❣♦♥❛❧ ✭♣❛r❛❧❡❧❛ à ❤✐♣♦t❡♥✉s❛✮ ❞♦ tr✐â♥❣✉❧♦✱ ❝♦♠❡ç❛♥❞♦ ♣❡❧♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ❞❛ ❞✐❛❣♦♥❛❧✱ é ✐❣✉❛❧ ❛♦ ❡❧❡♠❡♥t♦ q✉❡ ❡stá ✐♠❡❞✐❛t❛♠❡♥t❡ ❛❜❛✐①♦ ❞♦ ú❧t✐♠♦ ❝♦❡✜❝✐❡♥t❡ ❜✐♥♦♠✐❛❧ ❞❛ s♦♠❛✳
❋✐❣✉r❛ ✶✳✸✿ ❚❡♦r❡♠❛ ❞❛s ❞✐❛❣♦♥❛✐s
❊s❝r❡✈❡♥❞♦ s♦❜ ❛ ♥♦t❛çã♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❜✐♥♦♠✐❛✐s✱ t❡♠♦s✿
n 0 +
n+ 1 1
+
n+ 2 2
+· · ·+
n+p p
=
n+p+ 1
p
. ❏✉st✐✜❝❛t✐✈❛✿ ❚❡♠♦s✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦s ❜✐♥♦♠✐❛✐s ❝♦♠♣❧❡♠❡♥t❛r❡s✱ q✉❡✿
n 0 ❂ n n ❀
n+ 1 1
❂
n+ 1
n
❀
n+ 2 2
❂
n+ 2
n
❀· · ·❀
n+p p
❂
n+p n . ❉❡ss❡ ♠♦❞♦✿ n 0 +
n+ 1 1
+
n+ 2 2
+· · ·+
n+p p = n n +
n+ 1
n
+· · ·+
n+p n
.
❖ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦ ❈❛♣ít✉❧♦ ✶
P❡❧♦ ❚❡♦r❡♠❛ ❞❛s ❝♦❧✉♥❛s✱ s❛❜❡♠♦s q✉❡
n n
+
n+ 1
n
+
n+ 2
n
+
n+ 3
n
+· · ·+
n+p n
=
n+p+ 1
n+ 1
, ❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦s ❜✐♥♦♠✐❛✐s ❝♦♠♣❧❡♠❡♥t❛r❡s✱ t❡♠♦s
n+p+ 1
n+ 1
=
n+p+ 1
p
.
❆ss✐♠ ❝♦♥❝❧✉í♠♦s ❛ ❥✉st✐✜❝❛t✐✈❛ ❞♦ ❚❡♦r❡♠❛ ❞❛s ❞✐❛❣♦♥❛✐s✱ ♣r♦✈❛♥❞♦ q✉❡
n
0
+
n+ 1 1
+
n+ 2 2
+· · ·+
n+p p
=
n+p+ 1
p
,
❝♦♠♦ ❞❡s❡❥❛❞♦✳
Pr♦♣r✐❡❞❛❞❡ ✽ ✭❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ♥♦ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦✮✳ ✭❱❡❥❛ ❬✶✵❪✮ ❖ ♥ú♠❡r♦ ❞❡ ❋✐❜♦♥❛❝❝✐ Fné ♦❜t✐❞♦ ❝♦♠♦ ❛ s♦♠❛ ❞❛ ♥✲és✐♠❛ ✏❞✐❛❣♦♥❛❧ ✐♥✈❡rs❛✑
❞♦ ❚r✐â♥❣✉❧♦ ❞❡ ❆r✐t♠ét✐❝♦✳
❋✐❣✉r❛ ✶✳✹✿ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ♥♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦
❏✉st✐✜❝❛t✐✈❛✿ ❖❜s❡r✈❛♠♦s ♣❡❧❛ ✜❣✉r❛ ❞❛❞❛ q✉❡ ❛ s♦♠❛ ❞❛s ✏❞✐❛❣♦♥❛✐s ✐♥✈❡rs❛s✑ ❣❡r❛ ❛ s❡q✉ê♥❝✐❛{1,1,2,3,5,8,13,21, ...}✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ ❡st❛ é ❡①❛t❛♠❡♥t❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ q✉❡ é ❞❡✜♥✐❞❛ ❛tr❛✈és ❞❛ r❡❝♦rrê♥❝✐❛✿
F0 = F1 = 1
Fn+2 = Fn+1 + Fn ♣❛r❛ n ≥0.
P❛r❛ t❛♥t♦ ❞❡✈❡♠♦s ✈❡r✐✜❝❛r q✉❡ ❡st❛ s❡q✉ê♥❝✐❛ s❛t✐s❢❛③ ❡st❛ r❡❝♦rrê♥❝✐❛✳ ❈♦♠❡❝❡♠♦s ♦❜s❡r✈❛♥❞♦ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢♦r♠❛çã♦ ❞❛ s❡q✉ê♥❝✐❛✱ ♥♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✳
❱❡♠♦s q✉❡✿
F0 =
0 0
= 1
F1 =
1 0
= 1
F2 =
2 0 + 1 1
F3 =
3 0 + 2 1
F4 =
4 0 + 3 1 + 2 2 ✳✳✳ Fn =
n 0 +
n−1 1
+
n−2 2
+ · · · +
n−k k
Fn+1 =
n+ 1 0 + n 1 +
n−1 2
+ · · · +
n+ 1−p p
Fn+2 =
n+ 2 0
+
n+ 1 1 + n 2 + · · · +
n+ 2−s s
♦♥❞❡ k, p ❡ s sã♦ ♦s ♠❛✐♦r❡s ♥ú♠❡r♦s ✐♥t❡✐r♦s q✉❡ s❛t✐s❢❛③❡♠
k ≤ n−k p ≤ n+ 1−p s ≤ n+ 2−s
♦✉ s❡❥❛
k ≤ n/2
p ≤ (n+ 1)/2
s ≤ (n+ 2)/2 = (n/2) + 1.
◆♦t❡ q✉❡ q✉❛♥❞♦né í♠♣❛r t❡♠♦sk= (n−1)/2❡p= (n+1)/2 = (n−1+2)/2 = k+1✳
◗✉❛♥❞♦ né ♣❛r t❡♠♦s q✉❡k =n/2 =p✳ ❆✐♥❞❛✱s=k+ 1✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ♣❛r✐❞❛❞❡
❞❡n✳ ■r❡♠♦s ✈❡r✐✜❝❛r q✉❡ ❛ s♦♠❛ ❞❡Fn❡Fn+1éFn+2✳ ❈♦♥s✐❞❡r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡
♦ ❝❛s♦ ❡♠ q✉❡ n é í♠♣❛r✳
❖ ❚r✐â♥❣✉❧♦ ❆r✐t♠ét✐❝♦ ❈❛♣ít✉❧♦ ✶
◆❡st❡ ❝❛s♦ t❡♠♦s✿
Fn+1+Fn =
n+ 1 0 + n 1 +
n−1 2
+ · · · +
n+ 1−(k+ 1)
k+ 1
+ n 0 +
n−1 1
+
n−2 2
+ · · · +
n−k k
=
n+ 1 0 + n 0 + n 1 +
n−1 1
+
n−1 2
+ · · · +
n−k k
+
n−k k+ 1
.
❆♣❧✐❝❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❙t✐❢❡❧ ❛♦s ♥ú♠❡r♦s ❜✐♥♦♠✐❛✐s ❡♠ ❝❛❞❛ ❝♦❧❝❤❡t❡s ✈❡♠♦s q✉❡
Fn+1+Fn =
n+ 1 0
+
n+ 1 1 + n 2 + · · · +
n+ 1−k k+ 1
.
❯s❛♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ s = k+ 1 ❡ q✉❡ t♦❞♦ ♥ú♠❡r♦ ❜✐♥♦♠✐❛❧ ❝♦♠ ❞❡♥♦♠✐♥❛❞♦r 0
t❡♠ ♦ ♠❡s♠♦ ✈❛❧♦r✱ 1✱ ♦❜t❡♠♦s
Fn+1+Fn =
n+ 2 0
+
n+ 1 1 + n 2 + · · · +
n+ 2−s s
=Fn+2.
❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ♦ ❝❛s♦ ❡♠ q✉❡ n é ♣❛r✳ ❏á q✉❡ k = p✱ Fn+1 ❡ Fn t❡rã♦ ♦
♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ♣❛r❝❡❧❛s✳ ❊♥tã♦✱ ❝♦♠♦ ♥♦ ❝❛s♦ ❛♥t❡r✐♦r ♦❜t❡♠♦s
Fn+1+Fn =
n+ 1 0 + n 1 +
n−1 2
+ · · · +
n+ 1−k k + n 0 +
n−1 1
+ · · · +
n+ 1−k k−1
+
n−k k
=
n+ 1 0 + n 0 + n 1 +
n−1 1
+
n−1 2
+ · · ·+
n+ 1−k k−1
+
n+ 1−k k
+
n−k k
.
◆♦✈❛♠❡♥t❡ ❛♣❧✐❝❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❙t✐❢❡❧ ❛♦s ♥ú♠❡r♦s ❜✐♥♦♠✐❛✐s ❞♦s ❝♦❧❝❤❡t❡s ❝♦♥❝❧✉í♠♦s q✉❡
Fn+1+Fn =
n+ 1 0
+
n+ 1 1 + n 2 + · · · +
n+ 2−k k
+
n−k k
.
▲❡♠❜r❛♥❞♦ q✉❡2k=n ❡s =k+ 1♦❜t❡♠♦sn+ 2−k =n+ 2−(s−1)✱n−k =k ❡ n+ 2−s= 2k+ 2−s = 2s−s =s✳ ❉❛í
n+ 1 0
= 1 =
n+ 2 0
,
n+ 2−k k
=
n+ 2−(s−1)
s−1
❡
n−k k
=
k k
= 1 =
s s
=
n+ 2−s s
.
P♦rt❛♥t♦ Fn+1+Fn=Fn+2✱ t❛♠❜é♠ ♥♦ ❝❛s♦ ❡♠ q✉❡n é ♣❛r✳
✶✳✹ ❙♦♠❛tór✐♦
❯♠❛ ✐♠♣♦rt❛♥t❡ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❛ ➪❧❣❡❜r❛ é ♦ s♦♠❛tór✐♦ ✭q✉❡ é r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❧❡tr❛ ❣r❡❣❛ ♠❛✐ús❝✉❧❛ s✐❣♠❛✿ Σ✮✱ ♦ q✉❛❧✱ s❡❣✉♥❞♦ ❬✻❪✱ r❡♣r❡s❡♥t❛ ❛ s♦♠❛ ❞❡ ✉♠
❝❡rt♦ ♥ú♠❡r♦ ❞❡ ♣❛r❝❡❧❛s ❝♦♠ ❛❧❣✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❝♦♠✉♠✳
❖ s♦♠❛tór✐♦ é ✉t✐❧✐③❛❞♦ ❝♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ r❡♣r❡s❡♥t❛r ❞❡ ❢♦r♠❛ s✉❝✐♥t❛ ✉♠❛ s♦♠❛ ✜♥✐t❛ ♦✉ ✐♥✜♥✐t❛✱ q✉❡ ♣♦ss✉❛ ✉♠❛ ❡①♣r❡ssã♦ ❣❡r❛❧ ♣❛r❛ s✉❛ ❢♦r♠❛çã♦✳ ❊❧❡ s❡rá ✉t✐❧✐③❛❞♦ ♠❛✐s t❛r❞❡✱ q✉❛♥❞♦ ✜③❡r♠♦s ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ t❡♦r❡♠❛ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❜✐♥♦♠✐❛❧ ♣❛r❛ q✉❛❧q✉❡r ♣♦tê♥❝✐❛ ✐♥t❡✐r❛ ♣♦s✐t✐✈❛ ❞❡ n✳
P♦❞❡♠♦s r❡♣r❡s❡♥t❛r✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ an✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
n
X
i=1
ai =a1+a2+a3 +· · ·+an−1+an.
◆❡st❡ ❝❛s♦✱ t❡♠♦s q✉❡ ♦s ✈❛❧♦r❡s ❞❡ i ✈❛r✐❛♠✱ ❞❡s❞❡ i = 1 ✭q✉❡ ❝❤❛♠❛♠♦s ❞❡
✏í♥❞✐❝❡ ✐♥❢❡r✐♦r✑✮ ❛té i=n ✭q✉❡ ❝❤❛♠❛♠♦s ❞❡ ✏í♥❞✐❝❡ s✉♣❡r✐♦r✑✮✱ ❝♦♠i ♥❛t✉r❛❧✳ ❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❛♣❧✐❝❛çã♦ ❞❡ s♦♠❛tór✐♦s✿
❊①❡♠♣❧♦ ✶✿ ❈❛❧❝✉❧❡ ♦ ✈❛❧♦r ❞❡✿
❛✮
5
X
i=1
i2 ❜✮ 4
X
k=0
k! ❝✮
10
X
n=1
1
n −
1
n+ 1
❙♦♠❛tór✐♦ ❈❛♣ít✉❧♦ ✶ ❙♦❧✉çã♦✿ ❛✮❚❡♠♦s q✉❡ 5 X i=1
i2 ❂ 12+ 22+ 32+ 42+ 52 ❂1 + 4 + 9 + 16 + 25 ❂55
❜✮ ❱❛♠♦s s✉❜st✐t✉✐r✱ ♥❛ ❡①♣r❡ssã♦ ❞❛❞❛✱ ♦s ✈❛❧♦r❡s ♥❛t✉r❛✐s✱ ❞❡ k = 0 ❛té k = 4✳
❆ss✐♠✱ t❡r❡♠♦s✿
4
X
k=0
k! = 0! + 1! + 2! + 3! + 4!
❡ ♣❡❧♦ q✉❡ ❢♦✐ ❞❡✜♥✐❞♦ ❛♥t❡r✐♦r♠❡♥t❡ s♦❜r❡ ❢❛t♦r✐❛❧✱ t❡♠♦s✿
4
X
k=0
k! = 1 + 1 + 2 + 6 + 24 = 34.
❝✮ ❙✉❜st✐t✉✐♥❞♦✱ ♥❛ ❡①♣r❡ssã♦ ❞❛❞❛✱ ♦s ✈❛❧♦r❡s ♥❛t✉r❛✐s ♣❛r❛ n✱ ❞❡s❞❡ n= 1 ✭í♥❞✐❝❡
✐♥❢❡r✐♦r✮ ❛té n = 10 ✭í♥❞✐❝❡ s✉♣❡r✐♦r✮✱ t❡r❡♠♦s✿
10 X n=1 1 n − 1
n+ 1
=
1−1 2 + 1 2− 1 3 + 1 3 − 1 4 +· · ·+ 1 9− 1 10 + 1 10 − 1 11 . ❊❧✐♠✐♥❛♥❞♦ ♦s ♣❛rê♥t❡s❡s ❡ ❝❛♥❝❡❧❛♥❞♦ ❛s ♣❛r❝❡❧❛s ♦♣♦st❛s✱ ♦❜t❡♠♦s✿
10 X n=1 1 n − 1
n+ 1
= 1− 1 11 =
10 11.
❊①❡♠♣❧♦ ✷✿ ❈❛❧❝✉❧❡ ♦ ✈❛❧♦r ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s s♦♠❛s✿
❛✮ 7 X i=0 7 i ❜✮ 19 X n=0
n+ 1
n ❝✮ 10 X p=2 p 2 ❙♦❧✉çã♦✿
❛✮ P❡❧♦ ✏❚❡♦r❡♠❛ ❞❛s ❧✐♥❤❛s✑✱ t❡♠♦s q✉❡✿
7 X i=0 7 i = 7 0 + 7 1 + 7 2 + 7 3 +· · ·+ 7 7
❜✮ ❯t✐❧✐③❛♥❞♦ ♦ ✏❚❡♦r❡♠❛ ❞❛s ❞✐❛❣♦♥❛✐s✑ ❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦s ❜✐♥♦♠✐❛✐s ❝♦♠♣❧❡♠❡♥t❛r❡s✱ t❡♠♦s✿ 19 X n=0
n+ 1
n = 1 0 + 2 1 + 3 2 + 4 3 +· · ·+ 20 19 = 21 19 = 21 2
= 21.20
2 = 210.
❝✮ P❡❧♦ ✏❚❡♦r❡♠❛ ❞❛s ❝♦❧✉♥❛s✑✱ t❡♠♦s✿
10 X p=2 p 2 = 2 2 + 3 2 + 4 2 + 5 2 +· · ·+ 10 2 = 11 3
= 165.
✶✳✺ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡
(
a
+
b
)
n◆♦ ❝♦♠❡ç♦ ❞❡st❡ ❝❛♣ít✉❧♦ r❡✈✐s❛♠♦s ❛❧❣✉♥s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ❞❡(a+b)n✳ ❱❛♠♦s
❛❣♦r❛ ♦❜s❡r✈❛r ❛❧❣✉♠❛s r❡❣✉❧❛r✐❞❛❞❡s ❞❡ss❡s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✿
• (a+b)2 =a2+ 2ab+b2 −→ ✸ t❡r♠♦s✱ ♦s ❡①♣♦❡♥t❡s ❞❡a ❞❡❝r❡s❝❡♠ ❞❡ 2 ❛té
③❡r♦ ❡ ♦s ❡①♣♦❡♥t❡s ❞❡ b ❛✉♠❡♥t❛♠ ❞❡s❞❡ ③❡r♦ ❛té 2❀
• (a+b)3 =a3+ 3a2b+ 3ab2+b3 −→ ✹ t❡r♠♦s✱ ♦s ❡①♣♦❡♥t❡s ❞❡ a ❞❡❝r❡s❝❡♠
❞❡ 3❛té ③❡r♦ ❡ ♦s ❡①♣♦❡♥t❡s ❞❡ b ❛✉♠❡♥t❛♠ ❞❡s❞❡ ③❡r♦ ❛té3❀
• (a+b)4 = a4 + 4a3b + 6a2b2 + 4ab3 +b4 −→ ✺ t❡r♠♦s✱ ♦s ❡①♣♦❡♥t❡s ❞❡ a
❞❡❝r❡s❝❡♠ ❞❡ 4❛té ③❡r♦ ❡ ♦s ❡①♣♦❡♥t❡s ❞❡ b ❛✉♠❡♥t❛♠ ❞❡s❞❡ ③❡r♦ ❛té4✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❬✻❪✱ ❡ss❛s ♦❜s❡r✈❛çõ❡s s✉❣❡r❡♠ q✉❡✱ ♣❛r❛ ❛ ♣❛rt❡ ❧✐t❡r❛❧ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a+b)n✱ n ∈ N✱
■✳ anb0
|{z}
1♦ t❡r♠♦
❀ an−1b1
| {z }
2♦ t❡r♠♦
❀ an−2b2
| {z }
3♦ t❡r♠♦
❀· · · ❀ a1bn−1
| {z }
n✲ és✐♠♦ t❡r♠♦
❀ a0bn
|{z}
(n+1) ✲ és✐♠♦ t❡r♠♦
❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a+b)n ❈❛♣ít✉❧♦ ✶
■■✳ ♦s ❝♦❡✜❝✐❡♥t❡s q✉❡ ❛♣❛r❡❝❡♠ ♥♦s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ❛♥t❡r✐♦r❡s ❝♦rr❡s♣♦♥❞❡♠✱ ♦r❞❡♥❛❞❛♠❡♥t❡✱ às ❧✐♥❤❛s ❞♦ ❚r✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✿
• (a+b)1 ❂ 1a+ 1b −→ ❧✐♥❤❛ ✶✿ ✶ ✶
• (a+b)2 ❂ a2+ 2ab+b2 −→ ❧✐♥❤❛ ✷✿ ✶ ✷ ✶
• (a+b)3 ❂ a3+ 3a2b+ 3ab2+b3 −→ ❧✐♥❤❛ ✸✿ ✶ ✸ ✸ ✶
❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a+b)n✱
❜❛st❛ ❝♦♥s✐❞❡r❛r ❛ ❧✐♥❛ n ✭❧✐♥❤❛ ❞❡ ♥✉♠❡r❛❞♦r n✮ ❞♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✳
n
0
| {z }
❝♦❡❢✳ ❞♦ ✶♦t❡r♠♦
n
1
| {z }
❝♦❡❢✳ ❞♦ ✷♦ t❡r♠♦
· · · ·
n n−1
| {z }
❝♦❡❢✳ ❞♦ ♥✲és✐♠♦ t❡r♠♦
n n
| {z }
❝♦❡❢✳ ❞♦ ✭♥✰✶✮✲és✐♠♦ t❡r♠♦
▲❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦s ❝♦❡✜❝✐❡♥t❡s ♦❜t✐❞♦s ❡♠ ■■ ❡ ❛ ♣❛rt❡ ❧✐t❡r❛❧ ♦❜t✐❞❛ ❡♠ ■✱ ♣♦❞❡♠♦s✱ ❡♥✜♠ ❡♥✉♥❝✐❛r ♦ ❚❡♦r❡♠❛ ❞❡ ◆❡✇t♦♥ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡
(a+b)n✱ ♣❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ ♥❛t✉r❛❧ n✳
✶✳✺✳✶ ❚❡♦r❡♠❛ ❞❡ ◆❡✇t♦♥
❙❡❥❛♠ a ❡ b ♥ú♠❡r♦s r❡❛✐s ❡n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ❡♥tã♦
(a+b)n=
n
0
anb0+
n
1
an−1b1+
n
2
an−2b2+ · · · +
n n
a0bn. ❯t✐❧✐③❛♥❞♦ ❛ ♥♦t❛çã♦ ❞❡ s♦♠❛tór✐♦✱ t❡♠♦s q✉❡✿
(a+b)n=
n
X
p=0
n p
an−pbp. ✭✶✳✶✮
❯♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ❞❡ ◆❡✇t♦♥ é ❛ q✉❡ ❬✶✵❪ ❛♣r❡s❡♥t❛✿ ❏✉st✐✜❝❛t✐✈❛✿ ❚❡♠♦s
(a+b)n= (a+b)(a+b)(a+b)· · ·(a+b)
✉♠ ♣r♦❞✉t♦ ❞❡ n ❢❛t♦r❡s✳ ❈❛❞❛ t❡r♠♦ ❞♦ ♣r♦❞✉t♦ é ♦❜t✐❞♦ ❡s❝♦❧❤❡♥❞♦✲s❡ ❡♠ ❝❛❞❛ ♣❛rê♥t❡s❡ ✉♠ a ♦✉ ✉♠ b ❡ ♠✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ♦s ❡s❝♦❧❤✐❞♦s✳ P❛r❛ ❝❛❞❛ ✈❛❧♦r ❞❡ p✱
0 ≤ p ≤ n✱ s❡ ❡s❝♦❧❤❡r♠♦s b ❡♠ p ❞♦s ♣❛rê♥t❡s❡s✱ a s❡rá ❡s❝♦❧❤✐❞♦ ❡♠ n −p ❞♦s ♣❛rê♥t❡s❡s ❡ ♦ ♣r♦❞✉t♦ s❡rá ✐❣✉❛❧ ❛an−pbp✳ ■ss♦ ♣♦❞❡ s❡r ❢❡✐t♦ ❞❡
n p
♠♦❞♦s✳ ❊♥tã♦
(a+b)n é ✉♠❛ s♦♠❛ ♦♥❞❡ ❤á✱ ♣❛r❛ ❝❛❞❛ p ∈ {0,1,· · · , n}✱
n p
♣❛r❝❡❧❛s ✐❣✉❛✐s ❛ an−pbp✱ ✐st♦ é✱ ❡①❛t❛♠❡♥t❡ ❛ s♦♠❛ q✉❡ t❡♠♦s ❡♠ ✭✶✳✶✮✳
❖❜s❡r✈❛çã♦ ✐♠♣♦rt❛♥t❡✿ ❖ ❚❡♦r❡♠❛ ❞❡ ◆❡✇t♦♥ t❛♠❜é♠ é ✈á❧✐❞♦ ♣❛r❛ q✉❛♥❞♦ q✉✐s❡r♠♦s ♦❜t❡r ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a−b)n✳ ◆❡st❡ ❝❛s♦✱ ❜❛st❛ ♣❡r❝❡❜❡r q✉❡✿
(a−b)n = [a+ (−b)]n. ❉❡ss❡ ♠♦❞♦✱ t❡r❡♠♦s✿
[a+ (−b)]n=
n
0
an(−b)0+
n
1
an−1(−b)1+
n
2
an−2(−b)2+· · ·+
n n
a0(−b)n.
❊♠ ❝❛❞❛ ✉♠ ❞♦s t❡r♠♦s ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❛❝✐♠❛✱ t❡♠♦s ♣♦tê♥❝✐❛s ❞♦ t✐♣♦ (−b)p✱
♦♥❞❡✿
(−b)p =
bp, s❡ ♣ é ♣❛r
−bp, s❡ ♣ é í♠♣❛r✳
P❡r❝❡❜❛♠♦s q✉❡ ♦s s✐♥❛✐s ❞♦s t❡r♠♦s ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a−b)n s❡ ❛❧t❡r♥❛♠✱
❛ ♣❛rt✐r ❞♦ ♣r✐♠❡✐r♦ t❡r♠♦✱ q✉❡ é ♣♦s✐t✐✈♦✳ ❙❡♥❞♦ ❛ss✐♠ ♣♦❞❡♠♦s✱ t❛♠❜é♠✱ ❢❛③❡r ❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a−b)n✱ ✉t✐❧✐③❛♥❞♦ ❛ ♥♦t❛çã♦ ❞❡ s♦♠❛tór✐♦✱
♦❜t❡♥❞♦✿
(a−b)n =
n
X
p=0
(−1)p
n p
an−pbp. ✭✶✳✷✮
❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a+b)n ❈❛♣ít✉❧♦ ✶
❱❡❥❛♠♦s✱ ❛❣♦r❛✱ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❛♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ◆❡✇t♦♥ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❜✐♥ô♠✐♦s ❡ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦s ✈❛❧♦r❡s ❞❡ ❛❧❣✉♠❛s ❡①♣r❡ssõ❡s✳
❊①❡♠♣❧♦ ✶✿ ❯t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ◆❡✇t♦♥✱ ✈❛♠♦s ❞❡s❡♥✈♦❧✈❡r ♦s s❡❣✉✐♥t❡s ❜✐♥ô♠✐♦s✿
❛✮ (x+ 3)5 ❜✮
x2− 1
x
6
❙♦❧✉çõ❡s✿
❛✮ ❚❡♠♦s✱ ♣❛r❛ n= 5✿ (x+ 3)5 =
5 0
.x5.30 +
5 1
.x4.31+
5 2
.x3.32 +
5 3
.x2.33
+
5 4
.x1.34+
5 5
.x0.35
= 1.x5.1 + 5.x4.3 + 10.x3.9 + 10.x2.27 + 5.x.81 + 1.1.243
= x5+ 15x4+ 90x3+ 270x2+ 405x+ 243
❜✮ P❛r❛ n = 6✱ t❡♠♦s✿
x2− 1
x 6 = 6 0
.x6.
−x1
0 + 6 1
.x5.
−1x
1 + 6 2
.x4.
−x1
2 + + 6 3
.x3.
−1x
3 + 6 4
.x2.
−x1
4 + 6 5
.x1.
−x1
5 + + 6 6
.x0.
−1x
6
= x6−6.x5.1
x + 15.x
4. 1
x2 −20.x 3. 1
x3 + 15.x 2. 1
x4 −6.x.
1
x5 +
1
x6
= x6−6x4+ 15x2−20 + 15
x2 −
6
x4 +
1
x6.
❊①❡♠♣❧♦ ✷✿ ❯t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ◆❡✇t♦♥✱ ♣r♦✈❡ ♦ ✏❚❡♦r❡♠❛ ❞❛s ❧✐♥❤❛s✑ ❞♦ tr✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✱ ♦✉ s❡❥❛✿
n 0 + n 1 + n 2 + n 3 + n 4 +· · ·+ n n
❙♦❧✉çã♦✿
P❛r❛ ♣r♦✈❛r ♦ t❡♦r❡♠❛ ❞❛ ❧✐♥❤❛s✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡(a+b)n✱ ❢❛③❡♥❞♦
a=b = 1✳ ❉❡ss❛ ❢♦r♠❛ t❡♠♦s✿ 2n = (1 + 1)n=
=
n
0
.1n.10+
n
1
1n−1.11+· · ·+
n k
.1n−k.1k+· · ·+
n n
10.1n
❈♦♠♦ 1n= 1✱∀ n ∈N✱ ❡♥tã♦✱
2n =
n 0 + n 1 + n 2 + n 3 + n 4 +· · ·+ n n ,
❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳
❊①❡♠♣❧♦ ✸✿ ❈❛❧❝✉❧❛r ♦s ✈❛❧♦r❡s ❞❛ s❡❣✉✐♥t❡s s♦♠❛s✿ ❛✮ 8 X p=0 8 p . 3 5
8−p
. 7 5 p ❜✮ 739 X p=0
(−1)p.
739
p
.4739−p.5p
❝✮ X = (1,3)5+ 5.(1,3)4.(1,7) + 10.(1,3)3.(1,7)2+ 10.(1,3)2.(1,7)3
+5.(1,3).(1,7)4+ (1,7)5.
❙♦❧✉çõ❡s✿
❛✮ ❯t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ◆❡✇t♦♥ ✭s♦❜ ❛ ♥♦t❛çã♦ ❞❡ s♦♠❛tór✐♦✮ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a+b)n✱ t❡♠♦s✿
8 X p=0 8 p . 3 5
8−p
. 7 5 p = 3 5+ 7 5 8 = 10 5 8
= 28 = 256
❜✮ ❯t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ◆❡✇t♦♥ ✭s♦❜ ❛ ♥♦t❛çã♦ ❞❡ s♦♠❛tór✐♦✮ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ (a−b)n✱ t❡♠♦s✿
739
X
p=0
(−1)p.
739
p
.4739−p.5p = (4
−5)739 = (−1)739 =−1