等差数列
算術級数の公式は、算術級数 Sn の各項を初項 a0 で書き換えたものと、末尾の項 an で書き換えたもの和から 2Sn を求めることで得られる:
S n = a 0 + ( a 0 + d ) + ( a 0 + 2 d ) + ⋯ + ( a 0 + n d ) + S n = a n + ( a n − d ) + ( a n − 2 d ) + ⋯ + ( a n − n d ) 2 S n = ( a 0 + a n ) + ( a 0 + d + a n − d ) + ( a 0 + 2 d + a n − 2 d ) + ⋯ + ( a 0 + n d + a n − n d ) {\displaystyle {\begin{aligned}S_{n}&=\color {red}a_{0}\color {green}+(a_{0}+d)\color {blue}+(a_{0}+2d)\color {black}+\dotsb \color {magenta}+(a_{0}+nd)\\[5pt]{}+S_{n}&=\color {red}a_{n}\color {green}+(a_{n}-d)\color {blue}+(a_{n}-2d)\color {black}+\dotsb \color {magenta}+(a_{n}-nd)\\\hline 2S_{n}&=\color {red}(a_{0}+a_{n})\color {green}+(a_{0}{\bcancel {{}+d}}+a_{n}{\bcancel {{}-d}})\color {blue}+(a_{0}{\bcancel {{}+2d}}+a_{n}{\bcancel {{}-2d}})\color {black}+\dotsb \color {magenta}+(a_{0}{\bcancel {{}+nd}}+a_{n}{\bcancel {{}-nd}})\end{aligned}}}