総和
∑ i = 1 n ( 2 i − 1 ) = n 2 {\displaystyle \textstyle \sum \limits _{i=1}^{n}(2i-1)=n^{2}}
∑ i = 0 n x i = x n + 1 − 1 x − 1 {\displaystyle \textstyle \sum \limits _{i=0}^{n}x^{i}={\dfrac {x^{n+1}-1}{x-1}}} (または x ( 1 − x n ) 1 − x + 1 ( x ≠ 1 ) {\displaystyle {\frac {x(1-x^{n})}{1-x}}+1\quad (x\neq 1)} でもよい。)
∑ i = 0 ∞ x i = 1 1 − x ( | x | < 1 ) {\displaystyle \textstyle \sum \limits _{i=0}^{\infty }x^{i}={\dfrac {1}{1-x}}\quad \left(|x|<1\right)}
∑ i = 0 n ( n i ) = 2 n {\displaystyle \textstyle \sum \limits _{i=0}^{n}\displaystyle {n \choose i}=2^{n}}
∑ i = 0 n − 1 ( i k ) = ( n k + 1 ) {\displaystyle \textstyle \sum \limits _{i=0}^{n-1}\displaystyle {i \choose k}={n \choose k+1}}