Sò͘-Ha̍k Kui-La̍p-Hoat

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.

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

Tē 1 pō͘: n = 1 ê sî,

${\displaystyle 1={\frac {1(1+1)}{2}}}$  sêng-li̍p

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

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

Tē 3 pō͘: Án-ne n = m + 1 ê sî,

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

mā ē sêng-li̍p.

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

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