a)
A=$\frac{1}{3^2}$ - $\frac{1}{3^3}$ + ... + $\frac{1}{3^{50}}$ - $\frac{1}{3^{51}}$
=> 3A = $\frac{1}{3}$ - $\frac{1}{3^2}$ + $\frac{1}{3^3}$ - ...+ $\frac{1}{3^{49}}$ - $\frac{1}{3^{50}}$
=> 3A + A= ($\frac{1}{3^2}$ - $\frac{1}{3^3}$ + ... + $\frac{1}{3^{50}}$ - $\frac{1}{3^{51}}$) + ($\frac{1}{3}$ - $\frac{1}{3^2}$ + $\frac{1}{3^3}$ - ...+ $\frac{1}{3^{49}}$ - $\frac{1}{3^{50}}$ )
Rút gọn đc:
4A= $\frac{1}{3}$ - $\frac{1}{3^{51}}$
4A= $\frac{3^{51}}{3^{52}}$ - $\frac{3}{3^{52}}$ (Quy đồng mẫu)
4A= $\frac{3^{51} - 3}{3^{52}}$
=>A= $\frac{3^{51} - 3}{3^{52}}$ . $\frac{1}{4}$
A= $\frac{3^{51} - 3}{3^{52} .4}$
b) B=$\frac{1}{1.2.3}$ + $\frac{1}{2.3.4}$ +...+ $\frac{1}{a.(a+1).(a+2)}$
=>2B= $\frac{2}{1.2.3}$ + $\frac{2}{2.3.4}$ +...+ $\frac{2}{a.(a+1).(a+2)}$
=>2B=($\frac{1}{1.2}$ - $\frac{1}{2.3}$) + ($\frac{1}{2.3}$ - $\frac{1}{3.4}$) + ... + ($\frac{1}{a.(a+1)}$ - $\frac{1}{(a+1).(a+2)}$)
Rút gọn đc:
2B=$\frac{1}{2}$ - $\frac{1}{(a+1).(a+2)}$
2B=$\frac{(a+1).(a+2) - 2}{2.(a+1).(a+2)}$
=>B=$\frac{(a+1).(a+2) - 2}{4.(a+1).(a+2)}$
