Sò͘-ha̍k kui-la̍p-hoat

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Sò͘-ha̍k kui-la̍p-hoat (mathematical induction) sī 1 chióng chèng-bêng ê hong-hoat, tiāⁿ ēng lâi chèng-bêng bó͘-mih tîn-su̍t tùi só͘-ū ê chū-jiân-sò͘ (natural number) lóng sī chin--ê. Chit ê hong-hoat ē-tàng khok-chhiong chò kiat-kò͘ kui-la̍p-hoat (structural induction), ēng tiàm khah it-poaⁿ-te̍k, ū liông-ki koan-hē (Well-founded relation) ê kiat-kò͘, pí-lūn kóng chhiū-á (chi̍p-ha̍p-lūn). Sò͘-ha̍k-te̍k ê lô-chek ham tiān-naú kho-ha̍k mā lóng ū leh ēng kiat-kò͘ kui-la̍p-hoat. Sò͘-ha̍k kui-la̍p-hoat ham chiah-ê ū hû-ha̍p liông-sū goân-chek (well-ordering principle) ê hong-hoat tī lô-chek téng-koân lóng sio-siâng, in lóng sī lô-chek téng-kè (logical equivalence) ê hong-hoat.

[siu-kái]

Chún kóng lán beh sò͘-ha̍k kui-la̍p-hoat lâi chèng-bêng 1 + 2 + 3 + \cdots + n = \frac{n(n + 1)}{2}tùi só͘-ū ê chū-jiân-sò͘ n lóng sêng-li̍p.

Tē 1 pō͘: n = 1 ê sî,

1 = \frac{1(1 + 1)}{2} sêng-li̍p

Tē 2 pō͘: ká-siat n = m ê sî sêng-li̍p,

1 + 2 + \cdots + m = \frac{m(m + 1)}{2}

Tē 3 pō͘: Án-ne n = m + 1 ê sî,

 1 + 2 + \cdots + m + (m + 1)
= \frac{m(m + 1)}{2} + (m+ 1)
= \frac{m(m + 1)}{2} + \frac{2(m + 1)}{2}
= \frac{(m + 2)(m + 1)}{2}
= \frac{(m + 1)(m + 2)}{2}
= \frac{(m + 1)((m + 1) + 1)}{2}.

mā ē sêng-li̍p.

Kin-kì Sò͘-ha̍k kui-la̍p-hoat, 1 + 2 + 3 + \cdots + n = \frac{n(n + 1)}{2} tùi só͘-ū ê chū-jiân-sò͘ n lóng sêng-li̍p. #