sè ❜ë ✸ ❝â t❤➸ rót r❛ tø n ♣❤➛♥ tû ❦❤✐ ✤â t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ f3 (n) = Cn3 ✳ ❇➙② ❣✐í✱ t❛ ①❡♠ ①➨t ❤❛✐ t ợ n m tữỡ ự f3 (m + n) s➩ ❧➔ sè ❜ë ❜❛ ❝õ❛ t➟♣ A✱ ❝ë♥❣ ✈ỵ✐ sè ❜ë ❜❛ ❝õ❛ t➟♣ B ❝ë♥❣ ✈ỵ✐ ♠ët sè ❤↕♥❣ ❦➳t ❤đ♣ ❝õ❛ sè ❜ë ❜❛ ✈ỵ✐ ✈➔✐ ♣❤➛♥ tû ❝õ❛ ♠é✐ t➟♣ ✈➟② f2 (n) = f3 (m + n) = f3 (m) + f3 (n) + mf2 (n) + nf2 (m) = f3 (m) + f3 (n) + (mn2 + nm2 ) − mn ✣à♥❤ ♥❣❤➽❛ g3 : N → R ❜ð✐ n3 n g3 (n) = f3 (n) − + ✈ỵ✐ n ∈ N, t❛ ❝â g3 (m + n) = g3 (m) + g3 (n) ❉♦ ✤â f3 (n) = cn − n2 n3 + ❱➻ f3 (3) = 1, t❛ ❝â c= ✈➔ f3 (n) = n(n − 1)(n − 2) = Cn3 ✷✳✹✳ ❚ê♥❣ ❝õ❛ ❝❤✉é✐ ❤ú✉ ❤↕♥ ✭✐✮ ❈❤♦ S(n) = 1.2 + 2.3 + + n(n + 1) ✈ỵ✐ n ∈ N, ✭✷✳✶✹✮ ✹✺ tr♦♥❣ ✤â S : N → N ❞♦ ✤â S(m + n) = S(n) + S(m) + mn2 + nm2 + 2mn ❉♦ ✤â f (m + n) = f (m) + f (n), tr♦♥❣ ✤â n3 f (n) = S(n) − n − ✈ỵ✐ n ∈ N ✈➔ f : N → R✳ ❉♦ ✤â f ❧➔ ❝ë♥❣ t➼♥❤ ✈➔ f (n) = cn✳ ❱➟② S(n) = cn + n2 + n3 ❉♦ ✤â S(1) = 2✱ t❛ ❝â S(n) = n(n + 1)(n + 2) ✭✷✳✶✺✮ ✭✐✐✮ ❈❤♦ t(n) = 1.3 + 2.5 + + n(n + 2) ✈ỵ✐ n ∈ N, ✭✷✳✶✻✮ tr♦♥❣ ✤â t : N N ú ỵ r t(1) = 3✳ ❇➙② ❣✐í t(m + n) = t(n) + t(m) + mn2 + nm2 + 3nm ✈ỵ✐ m, n ∈ N ✣à♥❤ ♥❣❤➽❛ f : N → R ❜ð✐ f (n) = t(n) − n3 − n2 ✈ỵ✐ n ∈ N ◗✉❛♥ ❤➺ tr✉② t♦→♥ tr➯♥ trð t❤➔♥❤ f (m + n) = f (m) + f (n) ✈ỵ✐ m, n ∈ N ✣â ❧➔ f ❧➔ ❝ë♥❣ t➼♥❤ ✈➔ f (n) = cn ✈ỵ✐ n ∈ N✳ ❉♦ t(1) = t❛ ❝â t(n) = n(n + 1)(2n + 1) ✈ỵ✐ n ∈ N ✭✷✳✶✼✮ ✭✐✐✐✮ ❚ê♥❣ ❝õ❛ t➼❝❤ ❤é♥ t↕♣✳ s(n) = 1.2.3 + 2.3.4 + + n(n + 1)(n + 2), ✈ỵ✐ n ∈ N, ✭✷✳✶✽✮ ✹✻ ❱ỵ✐ s : N N ú ỵ r s(1) = t m, n ≥ t❛ ❝â s(n + m − 1) = s(n − 1) + n(n + 1)(n + 2) + {(n + 1)(n + 2)(n + 3) + · · · + (n + m − 1)(n + m)(n + m + 1)} = s(n − 1) + (n3 + 3n2 + 2n) + (m − 1)n3 +n2 (6 + · · · + 3m) + · · · + n{[1 · + · · · + (m − 1)m] +[1 · + · · · + (m − 1)(m + 1)] +[2 · + m(m + 1)]} + s(m − 1) = s(n − 1) + s(m − 1) + mn3 + 3n2 (1 + + · · · + m) +n{[1 · + · · · + (m − 1)m] +[1 · + · · · + (m − 1)(m + 1)][1 · + · · · + m(m + 1)]} m(m + 1) = s(n − 1) + s(m − 1) + mn3 + 3n2 (m − 1)m(m + 1) m(m − 1)(2m + 5) +n +n + m(m − 1)(2m + 5) (m − 1)m(m + 1) +n n m(m + 1)(m + 2) +n ✭sû ❞ö♥❣ ✤➥♥❣ t❤ù❝ ✭✷✳✶✺✮✱ ✭✷✳✶✻✮✮ 3 = s(n − 1) + s(m − 1) + mn3 + m2 n2 + m3 n 3 + n m + nm2 − mn 2 ữ trữợ f : N R ①→❝ ✤à♥❤ ♥❤÷ s❛✉ f (n) = s(n − 1) − n4 − n + n ✈ỵ✐ n ∈ N 4 ❈❤♦ f (m+n) = f (m)+f (n) ✭❝ë♥❣ t➼♥❤✮ ✈➔ f (n) = cn✳ ❙û ❞ö♥❣ s(1) = 6✱ t❛ ❝â s(n − 1) = [n4 + 2n3 − n2 − 2n]; ♥❣❤➽❛ ❧➔ s(n) = n(n + 1)(n + 2)(n + 3) ✈ỵ✐ n ∈ N ✭✷✳✶✾✮ ✹✼ ❑➳t ❧✉➟♥ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ ♠ët ❝❤õ ✤➲ ❧✐➯♥ q✉❛♥ tỵ✐ ❤➜✉ ❤➳t ❝→❝ ❦❤➼❛ ❝↕♥❤ ❝õ❛ ❚♦→♥ ❤å❝✳ ❚✉② ♥❤✐➯♥ ✈ỵ✐ ♣❤↕♠ ✈✐ ❝õ❛ ❧✉➟♥ ✈➠♥ ❚❤↕❝ s➽ ❝❤✉②➯♥ ♥❣➔♥❤ ♣❤÷ì♥❣ ♣❤→♣ ❚♦→♥ ❝➜♣ tỉ✐ t➟♣ tr✉♥❣ tr➻♥❤ ❜➔② ✈➲✿ ✧P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ù♥❣ ❞ư♥❣✧✳ ▲✉➟♥ ✈➠♥ ❝â ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ s❛✉✿ ✲ ❚r➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♠ơ✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❧♦❣❛r✐t✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥❤➙♥ t➼♥❤✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥❤✐➲✉ ❜✐➳♥ ✈➔ ♠ð rë♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ✲ ❚r➻♥❤ ❜➔② ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♥â tr➯♥ t➟♣ sè t❤ü❝ ✈➔ sè ♣❤ù❝✱ ❝❤➾ r❛ ♥❣❤✐➺♠ ❧✐➯♥ tö❝ ❝õ❛ ♥â ❧➔ t✉②➳♥ t➼♥❤✱ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ❤➔♠ sè ♠ô ❈❛✉❝❤② ♠➔ ❦❤ỉ♥❣ ❝➛♥ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ♥❤÷ ❧✐➯♥ tư❝✱ ❜à ❝❤➦♥ ❤❛② ❦❤↔ ✈✐✳ ◆❣❤✐➯♥ ❝ù✉ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥❤✐➲✉ ❜✐➳♥✳ ✲ ❚r➻♥❤ ❜➔② ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② tr♦♥❣ t➼♥❤ tê♥❣ ❧ơ② t❤ø❛ ❝õ❛ sè ♥❣✉②➯♥ ✭tê♥❣ ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥✱ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ n sè tü ♥❤✐➯♥ ✤➙✉ t✐➯♥✱ tê♥❣ ❧ô② t❤ø❛ k ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥✮✱ t➼♥❤ tê♥❣ ❧ô② t❤ø❛ ❝õ❛ ❝→❝ sè tr♦♥❣ ❞➣② ❝➜♣ sè ❝ë♥❣✱ t➻♠ sè ❝➦♣ ❝â t❤➸ rót r❛ tø n ♣❤➛♥ tû✱ ❧ü❝ ❧÷đ♥❣ ❝õ❛ ♠ët t➟♣ ❤ñ♣ ✈➔ tê♥❣ ❝õ❛ ❝❤✉é✐ ❤ú✉ ❤↕♥✳ ✹✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❆✳ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❚r➛♥ ✣ù❝ ▲♦♥❣✱ ❍♦➔♥❣ ◗✉è❝ ❚♦➔♥✱ ◆❣✉②➵♥ ✣➻♥❤ ❙❛♥❣ ✭✷✵✵✺✮✱ ●✐→♦ tr➻♥❤ ●✐↔✐ t➼❝❤✱ t➟♣ ✶✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍◆✳ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✭✶✾✾✼✮✱ P❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ◆❳❇ ●✐→♦ ❞ư❝✳ ❬✸❪ ◆❣✉②➵♥ ❱➠♥ ◆❤♦✱ ▲➯ ❍♦➔♥❣ P❤á ✭✷✵✶✸✮✱ ❚✉②➸♥ t➟♣ t t ữợ ❉÷ì♥❣✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍◆✳ ❇✳ ❚✐➳♥❣ ❆♥❤ ❬✹❪ ❏✳ ❆❝➨❧ ✭✶✾✻✻✮✱ ▲❡❝t✉r❡s ♦♥ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❚❤❡✐r ❛♣♣❧✐✲ ❝❛t✐♦♥s✳ ❬✺❪ ❈❤r✐st♦♣❤❡r ●✳ ❙♠❛❧❧ ✭✷✵✵✼✮✱ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❍♦✇ t♦ ❙♦❧✈❡ ❚❤❡♠✱ ❙♣r✐♥❣❡r✳ ❬✻❪ P✳ ❑❛♥♥❛♣♣❛♥ ✭✷✵✵✶✮✱ ✧❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❈❛✉❝❤②✬s ❊q✉❛t✐♦♥ ✐♥ ❈♦♠✲ ❜✐♥❛t♦r✐❝s ❛♥❞ ●❡♥❡t✐❝s✧✱ ▼❛t❤✇❛r❡ & ❙♦❢t ❈♦♠♣✉t✐♥❣✱ ✭✽✮✱ PP✳ ✻✶✲✻✹✳ ❬✼❪ P✳ ❑✳ ❙❛❤♦♦✱ P✳ ❑❛♥♥❛♣♣❛♥ ✭✷✵✶✶✮✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s✱ ❈❤❛♣♠❛♥ & ❍❛❧❧✴❈❘❈✳ ❬✽❪ ❙♦♦♥✲▼♦ ❏✉♥❣ ✭✷✵✶✵✮✱ ❍②❡rs✕❯❧❛♠✕❘❛ss✐❛s ❙t❛❜✐❧✐t② ♦❢ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s ✐♥ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ ❙♣r✐♥❣❡r✳ ❬✾❪ ❚✐t✉ ❆♥❞r❡❡s❝✉✱ ■✉r✐❡ ❇♦r❡✐❝♦ ✭✷✵✵✼✮✱ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s✱ ❊❧❡❝✲ tr♦♥✐❝ ❊❞✐t✐♦♥✳