等比数列
等比数列の初項から第 n 項までの和は以下の式で定義される。
∑ k = 0 n a r k = a + a r + ⋯ + a r n {\displaystyle \textstyle \sum \limits _{k=0}^{n}ar^{k}=a+ar+\dotsb +ar^{n}}
r ≠ 1 の場合、(1 − r) を掛けると、
( 1 − r ) ∑ k = 0 n a r k = a ( 1 − r ) ∑ k = 0 n r k = a ( 1 − r ) ( 1 + r + ⋯ + r n − 1 + r n ) = a ( ( 1 + r + ⋯ + r n − 1 + r n ) − ( r + r 2 + ⋯ + r n + r n + 1 ) ) = a ( 1 − r n + 1 ) {\displaystyle {\begin{aligned}(1-r)\textstyle \sum \limits _{k=0}^{n}ar^{k}&=a(1-r)\textstyle \sum \limits _{k=0}^{n}r^{k}\\&=a(1-r)(1+r+\dotsb +r^{n-1}+r^{n})\\&=a\left((1+{\bcancel {r+\dotsb +r^{n-1}+r^{n}}})\right.\\&\left.-({\bcancel {r+r^{2}+\dotsb +r^{n}}}+r^{n+1})\right)\\&=a(1-r^{n+1})\end{aligned}}}