- [tex]S_{1}= C_{n}^{1}+ 2.C_{n}^{2}+...+(n-1).C_{n}^{n-1}+n.C_{n}^{n}[/tex]
- [tex]S_{2}= C_{n}^{0}+2.C_{n}^{1}+...+n.C_{n}^{n-1}+(n+1).C_{n}^{n}[/tex]
- [tex]S_{3}=2.C_{n}^{0} + 5.C_{n}^{1} + 8.C_{n}^{3} +...+ (3n+2).C_{n}^{n}[/tex]
- [tex]S_{4}= 100.C_{100}^{0}.(\frac{1}{2})^{99} - 101.C_{100}^{1}.(\frac{1}{2})^{100} +...- 199.C_{100}^{99}.(\frac{1}{2})^{198} + 200.C_{100}^{100}.(\frac{1}{2})^{199}[/tex]
1) xét khai triển [tex](1+x)^n=C_n^0+xC_n^1+x^2 C_n^2+...+x^nC_n^n[/tex]
Đạo hàm bậc 1 ta được
[tex]n(1+x)^{n-1}=C_n^1+2xC_n^2+3x^2C_n^3+...+nx^{n-1}C_n^n[/tex]
Với $x=1$ [tex]\Rightarrow S_1=n.2^{n-1}[/tex]