Tính nhanh S = $\frac{2}{3^1}$ – $\frac{2}{3^2}$ + $\frac{2}{3^3}$ – $\frac{2}{3^4}$ + …+ $\frac{2}{3^89}$ – $\frac{2}{3^ 90}$

Đáp án:

$\begin{array}{l}
S = \dfrac{2}{{{3^1}}} – \dfrac{2}{{{3^2}}} + \dfrac{2}{{{3^3}}} – \dfrac{2}{{{3^4}}} + .. + \dfrac{2}{{{3^{89}}}} – \dfrac{2}{{{3^{90}}}}\\
 = 2.\left( {\dfrac{1}{{{3^1}}} – \dfrac{1}{{{3^2}}} + \dfrac{1}{{{3^3}}} – \dfrac{1}{{{3^4}}} + … + \dfrac{1}{{{3^{89}}}} – \dfrac{1}{{{3^{90}}}}} \right)\\
 = 2.A\\
A = \dfrac{1}{{{3^1}}} – \dfrac{1}{{{3^2}}} + \dfrac{1}{{{3^3}}} – \dfrac{1}{{{3^4}}} + … + \dfrac{1}{{{3^{89}}}} – \dfrac{1}{{{3^{90}}}}\\
 \Rightarrow 3.A = 1 – \dfrac{1}{{{3^1}}} + \dfrac{1}{{{3^2}}} – \dfrac{1}{{{3^3}}} + … + \dfrac{1}{{{3^{88}}}} – \dfrac{1}{{{3^{89}}}}\\
 \Rightarrow 3A + A = 4A = 1 – \dfrac{1}{{{3^{90}}}}\\
 \Rightarrow A = \dfrac{1}{4} – \dfrac{1}{{{{4.3}^{90}}}}\\
 \Rightarrow S = 2.\left( {\dfrac{1}{4} – \dfrac{1}{{{{4.3}^{90}}}}} \right)\\
 = \dfrac{1}{2} – \dfrac{1}{{{{2.3}^{90}}}}
\end{array}$