数学B シグマ公式と数列

シグマ公式

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

\sum_{k=1}^{n}k^2=\frac{1}{6}n(n+1)(2n+1)

\sum_{k=1}^{n}k^3=\Big\{\frac{1}{2}n(n+1)\Big\}^2

\sum_{k=1}^{n}r^{k-1}=\frac{r^n-1}{r-1}=\frac{1-r^n}{1-r}

a_n=a+(n-1)d=dn+a-d

\begin{aligned} \sum_{k=1}^{n}a_k&=\sum_{k=1}^{n}(dk+a-d) \\&=d\sum_{k=1}^{n}k+n(a-d) \\&=d\times\frac{1}{2}n(n+1)+n(a-d) \\&=\frac{n}{2}(2a+(n-1)d) \end{aligned}

b_n=b\times r^{n-1}

\begin{aligned} \sum_{k=1}^{n}b_k&=\sum_{k=1}^{n}(b\times r^{k-1}) \\&=b\sum_{k=1}^{n}r^{k-1} \\&=\frac{b(r^n-1)}{r-1} \end{aligned}