\(1,S=3+3^2+3^3+...+3^{20}\)(1)
\(\Rightarrow3S=3^2+3^3+3^4+...+3^{21}\)(2)
Lấy (2) -(1) ta có :
\(\Rightarrow2S=3^{21}-3\)
\(\Rightarrow S=\frac{3^{21}-3}{2}\)
\(3,A=1.2.3+2.3.4+3.4.5+...+\left(n-1\right)n\left(n+1\right)\)
\(\Rightarrow4A=1.2.3.4+2.3.4.\left(5-1\right)+3.4.5.\left(6-2\right)+...+\left(n-1\right)n\left(n+1\right)\left[\left(n+2\right)-\left(n-2\right)\right]\)
\(\Rightarrow4A=1.2.3.4+2.3.4.5-1.2.3.4+...+\left(n-1\right)n\left(n+1\right)\left(n+2\right)-\left(n-2\right)\left(n-1\right)n\left(n+1\right)\)
\(\Rightarrow4A=\left(n-1\right)n\left(n+1\right)\left(n+2\right)\)
\(\Rightarrow A=\frac{\left(n-1\right)n\left(n+1\right)\left(n+2\right)}{4}\)
