ITT9131 Konkr eetne Mate maatika Concrete

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■❚❚✾✶✸✶ ❑♦♥❦r❡❡t♥❡ ▼❛t❡♠❛❛t✐❦❛

❈♦♥❝r❡t❡ ♠❛t❤❡♠❛t✐❝s

Konkretna matematika

❏❛❛♥ P❡♥❥❛♠ ❙✐❧✈✐♦ ❈❛♣♦❜✐❛♥❝♦

❚❚Ü ❛r✈✉t✐t❡❛❞✉s❡ ✐♥st✐t✉✉t✱ t❡♦r❡❡t✐❧✐s❡ ✐♥❢♦r♠❛❛t✐❦❛ õ♣♣❡t♦♦❧

❑ü❜❡r♥❡❡t✐❦❛ ■♥st✐t✉✉❞✐ t❛r❦✈❛r❛t❡❛❞✉s❡ ❧❛❜♦r❛t♦♦r✐✉♠

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❚❤❡ ❜♦♦❦

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❈♦♥❝r❡t❡ ▼❛t❤❡♠❛t✐❝s ✐s ✳✳✳

t❤❡ ❝♦♥tr♦❧❧❡❞ ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛s ✉s✐♥❣ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ t❡❝❤♥✐q✉❡s ❢♦r s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ●♦❛❧s♦❢ t❤❡ ❜♦♦❦✿

t♦ ✐♥tr♦❞✉❝❡ t❤❡ ♠❛t❤❡♠❛t✐❝s t❤❛t s✉♣♣♦rts ❛❞✈❛♥❝❡❞ ❝♦♠♣✉t❡r ♣r♦❣r❛♠♠✐♥❣ ❛♥❞ t❤❡ ❛♥❛❧②s✐s ♦❢ ❛❧❣♦r✐t❤♠s t♦ ♣r♦✈✐❞❡ ❛ s♦❧✐❞ ❛♥❞ r❡❧❡✈❛♥t ❜❛s❡ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ s❦✐❧❧s ✲ t❤❡ s❦✐❧❧s ♥❡❡❞❡❞

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❖✉r ❛❞❞✐t✐♦♥❛❧ ❣♦❛❧s

t♦ ❣❡t ❛❝q✉❛✐♥t❡❞ ✇✐t❤ ✇❡❧❧✲❦♥♦✇♥ ❛♥❞ ♣♦♣✉❧❛r ❧✐t❡r❛t✉r❡ ✐♥ ❈❙ ❛♥❞ ▼❛t❤

t♦ ❞❡✈❡❧♦♣ ♠❛t❤❡♠❛t✐❝❛❧ s❦✐❧❧s✱ ❢♦r♠✉❧❛t✐♥❣ ❝♦♠♣❧❡① ♣r♦❜❧❡♠s ♠❛t❤❡♠❛t✐❝❛❧❧②

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❈♦♥t❡♥ts ♦❢ t❤❡ ❇♦♦❦

❈❤❛♣t❡rs✿

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s ✷ ❙✉♠s

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◆❡①t s❡❝t✐♦♥

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s

✷ ❙✉♠s

✸ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s

✹ ◆✉♠❜❡r ❚❤❡♦r②

✺ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

✻ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs

✼ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s

✽ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

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✶✳ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s

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◆❡①t s❡❝t✐♦♥

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s

✷ ❙✉♠s

✸ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s

✹ ◆✉♠❜❡r ❚❤❡♦r②

✺ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

✻ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs

✼ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s

✽ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

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✷✳ ❙✉♠s

✶ ◆♦t❛t✐♦♥

✷ ❙✉♠s ❛♥❞ ❘❡❝✉rr❡♥❝❡s ✸ ▼❛♥✐♣✉❧❛t✐♦♥ ♦❢ ❙✉♠s ✹ ▼✉❧t✐♣❧❡ ❙✉♠s

✺ ●❡♥❡r❛❧ ▼❡t❤♦❞s

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◆❡①t s❡❝t✐♦♥

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s

✷ ❙✉♠s

✸ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s

✹ ◆✉♠❜❡r ❚❤❡♦r②

✺ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

✻ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs

✼ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s

✽ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

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✸✳ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s

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◆❡①t s❡❝t✐♦♥

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s

✷ ❙✉♠s

✸ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s

✹ ◆✉♠❜❡r ❚❤❡♦r②

✺ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

✻ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs

✼ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s

✽ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

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✹✳ ◆✉♠❜❡r ❚❤❡♦r②

✶ ❉✐✈✐s✐❜✐❧✐t② ✷ ❋❛❝t♦r✐❛❧ ❋❛❝t♦rs ✸ ❘❡❧❛t✐✈❡ Pr✐♠❛❧✐t②

✹ ✬♠♦❞✬✿ ❚❤❡ ❈♦♥❣r✉❡♥❝❡ ❘❡❧❛t✐♦♥ ✺ ■♥❞❡♣❡♥❞❡♥t ❘❡s✐❞✉❡s

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◆❡①t s❡❝t✐♦♥

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s

✷ ❙✉♠s

✸ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s

✹ ◆✉♠❜❡r ❚❤❡♦r②

✺ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

✻ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs

✼ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s

✽ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

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✺✳ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

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◆❡①t s❡❝t✐♦♥

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s

✷ ❙✉♠s

✸ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s

✹ ◆✉♠❜❡r ❚❤❡♦r②

✺ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

✻ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs

✼ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s

✽ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

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✻✳ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs

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◆❡①t s❡❝t✐♦♥

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s

✷ ❙✉♠s

✸ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s

✹ ◆✉♠❜❡r ❚❤❡♦r②

✺ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

✻ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs

✼ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s

✽ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

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✼✳ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s

✶ ❉♦♠✐♥♦ ❚❤❡♦r② ❛♥❞ ❈❤❛♥❣❡ ✷ ❇❛s✐❝ ▼❛♥❡✉✈❡rs

✸ ❙♦❧✈✐♥❣ ❘❡❝✉rr❡♥❝❡s

✹ ❙♣❡❝✐❛❧ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s ✺ ❈♦♥✈♦❧✉t✐♦♥s

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◆❡①t s❡❝t✐♦♥

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s

✷ ❙✉♠s

✸ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s

✹ ◆✉♠❜❡r ❚❤❡♦r②

✺ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

✻ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs

✼ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s

✽ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

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✽✳ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

✶ ❉❡✜♥✐t✐♦♥s

✷ ▼❡❛♥ ❛♥❞ ❱❛r✐❛♥❝❡

✸ Pr♦❜❛❜✐❧✐t② ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s ✹ ❋❧✐♣♣✐♥❣ ❈♦✐♥s

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◆❡①t s❡❝t✐♦♥

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s

✷ ❙✉♠s

✸ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s

✹ ◆✉♠❜❡r ❚❤❡♦r②

✺ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

✻ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs

✼ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s

✽ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

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✾✳ ❆s②♠♣t♦t✐❝s

✶ ❆ ❍✐❡r❛r❝❤② ✷ O ◆♦t❛t✐♦♥

✸ O ▼❛♥✐♣✉❧❛t✐♦♥

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P❡❞❛❣♦❣✐❝❛❧ ❞✐❧❡♠♠❛✿ ✇❤❛t t♦ t❡❛❝❤❄

❈❤❛♣t❡rs✿

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s ✷ ❙✉♠s

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❘❡❝✉rr❡♥t ♣r♦❜❧❡♠s

❘❡❝✉rr❡♥t ❡q✉❛t✐♦♥

❆ ♥✉♠❜❡r s❡q✉❡♥❝❡hani=ha✵,a✶,a✷, . . .i ✐s ❝❛❧❧❡❞r❡❝✉rr❡♥t✱ ✐❢ ✐ts ❣❡♥❡r❛❧ t❡r♠ t♦❣❡t❤❡r ✇✐t❤k ♣r❡❝❡❞✐♥❣ t❡r♠s s❛t✐s✜❡s ❛r❡❝✉rr❡♥t ❡q✉❛t✐♦♥

an=g(an−✶, . . . ,an₋k),

❢♦r ❡✈❡r②n>k✱ ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ g:Nk _−→N✭♦rg :Rk_−→R✮✳ ❚❤❡ ❝♦♥st❛♥tk ✐s ❝❛❧❧❡❞♦r❞❡r ♦❢ t❤❡ r❡❝✉rr❡♥t ❡q✉❛t✐♦♥ ✭♦r

❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✮✳

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❘❡❣✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❞❡✜♥❡❞ ❜② ❧✐♥❡s

Q_✵=✶

Q_✷=✹

Q_✶=✷

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❘❡❣✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❞❡✜♥❡❞ ❜② ❧✐♥❡s

❆❝t✉❛❧❧② ✳✳✳

V_✶

V_✷

V_✸

V_✹

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❘❡❣✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❞❡✜♥❡❞ ❜② ❧✐♥❡s

❆❝t✉❛❧❧② ✳✳✳

V_✶✶ V_✶✷

V_✷✶

V_✷✷

V_✸

V_✹✶

V_✹✷

Q_✸=Q_✷+✸=✼

●❡♥❡r❛❧❧②Qn=Qn−✶+n✳

n ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ _···

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❘❡❣✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❞❡✜♥❡❞ ❜② ❧✐♥❡s

❆❝t✉❛❧❧② ✳✳✳

V_✶✶ V_✶✷

V_✷✶

V_✷✷

V_✸

V_✹✶

V_✹✷

Q_✸=Q_✷+✸=✼

●❡♥❡r❛❧❧②Qn=Qn−✶+n✳

n ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ _···

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❘❡❣✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❞❡✜♥❡❞ ❜② ❧✐♥❡s

T_✵=✶

T_✷=✼

T_✶=✷

T_✸=❄

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❘❡❣✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❞❡✜♥❡❞ ❜② ❧✐♥❡s

T_✷ = Q_✹₋✷_·✷=✶✶−✹=✼

T_✸ = Q_✻₋✷_·✸=✷✷−✻=✶✻

T_✹ = Q_✽−✷·✹=✸✼−✽=✷✾

T_✺ = Q_✶✵₋✷_·✺=✺✻−✶✵=✹✻

Tn = Q✷n−✷n

n ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ _···

Qn ✶ ✷ ✹ ✼ ✶✶ ✶✻ ✷✷ ✷✾ ✸✼ ✹✻ ···

(34)

❘❡❣✐♦♥s ♦❢ t❤❡ ♣❧❛♥❡ ❞❡✜♥❡❞ ❜② ❧✐♥❡s

T_✷ = Q_✹₋✷_·✷=✶✶−✹=✼

T_✸ = Q_✻₋✷_·✸=✷✷−✻=✶✻

T_✹ = Q_✽−✷·✹=✸✼−✽=✷✾

T_✺ = Q_✶✵₋✷_·✺=✺✻−✶✵=✹✻

Tn = Q✷n−✷n

n ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ _···

(35)

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(41)

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(46)

(47)

P♦ss✐❜❧❡ s❝❤❡❞✉❧❡

❈❤❛♣t❡rs✿

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s✕ ✇❡❡❦s ✶✕✸ ✷ ❙✉♠s ✕ ✇❡❡❦s ✹✕✻

✸ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s ✕ ✇❡❡❦ ✼ ✹ ◆✉♠❜❡r ❚❤❡♦r②

✺ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

✻ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs✕ ✇❡❡❦s ✽✕✾ ✼ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s✕ ✇❡❡❦s ✶✵✕✶✸ ✽ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

(48)

P♦ss✐❜❧❡ s❝❤❡❞✉❧❡

❈❤❛♣t❡rs✿

✶ ❘❡❝✉rr❡♥t Pr♦❜❧❡♠s✕ ✇❡❡❦s ✶✕✸ ✷ ❙✉♠s ✕ ✇❡❡❦s ✹✕✻

✸ ■♥t❡❣❡r ❋✉♥❝t✐♦♥s ✕ ✇❡❡❦ ✼ ✹ ◆✉♠❜❡r ❚❤❡♦r②

✺ ❇✐♥♦♠✐❛❧ ❈♦❡✣❝✐❡♥ts

✻ ❙♣❡❝✐❛❧ ◆✉♠❜❡rs✕ ✇❡❡❦s ✽✕✾ ✼ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s✕ ✇❡❡❦s ✶✵✕✶✸ ✽ ❉✐s❝r❡t❡ Pr♦❜❛❜✐❧✐t②

✾ ❆s②♠♣t♦t✐❝s

(49)

❈♦♥t❛❝t

■♥str✉❝t♦rs✿ ❙✐❧✈✐♦ ❈❛♣♦❜✐❛♥❝♦ ❏❛❛♥ P❡♥❥❛♠ ❆❞❞r❡ss❡s✿ s✐❧✈✐♦❅❝s✳✐♦❝✳❡❡ ❥❛❛♥❅❝s✳✐♦❝✳❡❡

▲❡❝t✉r❡s ❛♥❞ ❡①❡r❝✐s❡s✿ ❖♥ ❚✉❡s❞❛②s ❛t ✶✹✿✵✵ ✕ ✶✼✿✸✵ ✐♥ ❇✶✷✻

❲❡❜ ♣❛❣❡✿