$ \sum{ k}\;的公式 $

$ Pf： $

$ 設S=\sum_{k=1}^{n}{k } $

$ S=1\;+2\;+3\;+...+(n-2)+(n-1)+n $

$ S=n+(n-2)+(n-1)+..+3+2+1 $

$ 2S=(n+1)+(n+1)+...+(n+1)=n(n+1) $

$ S=\sum_{k=1}^{n}{k }=\frac{n(n+1)}{2} $

n = 1 (slider from 1 to 50, step 1)
text1: "$ \sum_{k=1}^{n}{k }=\frac{n(n+1)}{2} $
$\sum_{k=1}^{" + (LaTeX[n]) + "}{ k}=" + (LaTeX[If[n ≟ 1, "1", If[n ≟ 2, "1+2", If[n ≟ 3, "1+2+3", "1+2+..."]]]]) + "" + (LaTeX[If[n < 4, "", "+"]]) + "" + (LaTeX[If[n > 3, n]]) + "$
$\;\;\;\;\;\;\;\;=\frac{" + (LaTeX[n]) + "×" + (LaTeX[n + 1]) + "}{2}$
$\;\;\;\;\;\;\;\;=" + (LaTeX[(n * (n + 1)) / 2]) + "$"
Step = 0 (slider from 0 to 5, step 1)
A = (1, 5.64)