Biết rằng S = 1 + 2.3 + 3.3^2 + ... + 11.3^10 = a + 21.3^b/4. Tính P = a + b/4.

Từ giả thiết suy ra \[3S = 3 + {2.3^2} + {3.3^3} + ... + {11.3^{11}}\]. Do đó

\( - 2S = S - 3S = 1 + 3 + {3^2} + ... + {3^{10}} - {10.3^{11}} = \frac{{1 - {3^{11}}}}{{1 - 3}} - {11.3^{11}} = - \frac{1}{2} - \frac{{{{21.3}^{11}}}}{2} \Rightarrow S = \frac{1}{4} + \frac{{21}}{4}{.3^{11}}.\)                  

Vì $S=\frac{1}{4}+\frac{{21.3}^{11}}{4}=a+\frac{{21.3}^b}{4}\Rightarrow a=\frac{1}{4},b=11\underset{}{\to }P=\frac{1}{4}+\frac{11}{4}=3.$