Tính nhanh
S = $\frac{2}{3^1}$ – $\frac{2}{3^2}$ + $\frac{2}{3^3}$ – $\frac{2}{3^4}$ + …+ $\frac{2}{3^89}$ – $\frac{2}{3^ 90}$
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: `S = \frac{3^{90}-1}{3^{90}.2}`.
Giải thích các bước giải:
`S = 2/3 – 2/{3^2} + 2/{3^3} – 2/{3^4} + …. + 2/{3^{89}} – 2/{3^{90}}`
`⇔ 3S = 2 – 2/3 + 2/{3^2} – 2/{3^3} + …. + 2/{8^{88}} – 2/{3^{89}}`
`⇔ 3S + S = (2 – 2/3 + 2/{3^2} – 2/{3^3} + …. + 2/{8^{88}} – 2/{3^{89}})+( 2/3 – 2/{3^2} + 2/{3^3} – 2/{3^4} + …. + 2/{3^{89}} – 2/{3^{90}})`
`⇔ 4S = 2 – 2/{3^{90}}`
`⇔ 4S = {2.3^{90} – 2}/{3^{90}}`
`⇔ S = \frac{2(3^{90}-1)}{3^{90}.4}`
`⇔ S = \frac{3^{90}-1}{3^{90}.2}`.
Đáp án:
$\begin{array}{l}
S = \dfrac{2}{{{3^1}}} – \dfrac{2}{{{3^2}}} + \dfrac{2}{{{3^3}}} – \dfrac{2}{{{3^4}}} + .. + \dfrac{2}{{{3^{89}}}} – \dfrac{2}{{{3^{90}}}}\\
= 2.\left( {\dfrac{1}{{{3^1}}} – \dfrac{1}{{{3^2}}} + \dfrac{1}{{{3^3}}} – \dfrac{1}{{{3^4}}} + … + \dfrac{1}{{{3^{89}}}} – \dfrac{1}{{{3^{90}}}}} \right)\\
= 2.A\\
A = \dfrac{1}{{{3^1}}} – \dfrac{1}{{{3^2}}} + \dfrac{1}{{{3^3}}} – \dfrac{1}{{{3^4}}} + … + \dfrac{1}{{{3^{89}}}} – \dfrac{1}{{{3^{90}}}}\\
\Rightarrow 3.A = 1 – \dfrac{1}{{{3^1}}} + \dfrac{1}{{{3^2}}} – \dfrac{1}{{{3^3}}} + … + \dfrac{1}{{{3^{88}}}} – \dfrac{1}{{{3^{89}}}}\\
\Rightarrow 3A + A = 4A = 1 – \dfrac{1}{{{3^{90}}}}\\
\Rightarrow A = \dfrac{1}{4} – \dfrac{1}{{{{4.3}^{90}}}}\\
\Rightarrow S = 2.\left( {\dfrac{1}{4} – \dfrac{1}{{{{4.3}^{90}}}}} \right)\\
= \dfrac{1}{2} – \dfrac{1}{{{{2.3}^{90}}}}
\end{array}$