\(\sum\)の計算
【\(\sum\)の定義】
\(\displaystyle \sum_{k=1}^{n}a_k=a_1+a_2+a_3+\cdots+a_n\)【\(\sum\)の公式】
(1)\(\displaystyle \sum_{k=1}^{n}c=cn\ \ c\)は定数(2)\(\displaystyle \sum_{k=1}^{n}k=\frac{1}{2}n(n+1)\)
(3)\(\displaystyle \sum_{k=1}^{n}k^2=\frac{1}{6}n(n+1)(2n+1)\)
(4)\(\displaystyle \sum_{k=1}^{n}k^3=\left\{\frac{1}{2}n(n+1)\right\}^2\)
(5)\(\displaystyle \sum_{k=1}^{n}ar^{k-1}=\frac{a(1-r^n)}{1-r}\ \ r\neq1\)は定数
【\(\sum\)の性質】
(1)\(\displaystyle \sum_{k=1}^{n}(a_k+b_k)=\sum_{k=1}^{n}a_k+\sum_{k=1}^{n}b_k\)(2)\(\displaystyle \sum_{k=1}^{n}ca_k=c\sum_{k=1}^{n}a_k\ \ c\)は定数
【例題】次の数列の和を求めなさい。
(1)\(\displaystyle \sum_{k=1}^{15}2\)
\(=15・2\)
\(=30\)
(2)\(\displaystyle \sum_{k=1}^{34}k\)
\(\displaystyle =\frac{1}{2}・34・35\)
\(=595\)
(3)\(\displaystyle \sum_{k=1}^{9}k^2\)
\(\displaystyle =\frac{1}{6}・9・10・19\)
\(=285\)
(4)\(\displaystyle \sum_{k=1}^{n}(8k-3)\)
\(\displaystyle =8・\frac{1}{2}n(n+1)-3n\)
\(=4n^2+4n-3n\)
\(=n(4n+1)\)
(5)\(\displaystyle \sum_{k=1}^{n-1}5k\)
\(\displaystyle =5・\frac{1}{2}(n-1)(n+1-1)\)
\(\displaystyle =\frac{5}{2}n(n-1)\)
(6)\(\displaystyle \sum_{k=1}^{n}7・2^{k-1}\)
\(\displaystyle =\frac{7(2^n-1)}{2-1}\)
\(=7(2^n-1)\)
(7)\(\displaystyle \sum_{k=1}^{n}\left(\frac{1}{3}\right)^{k-1}\)
\(\displaystyle =\frac{(1-(\frac{1}{3})^n)}{1-\frac{1}{3}}\)
\(\displaystyle =\frac{3}{2}\left\{1-\left(\frac{1}{3}\right)^n\right\}\)