so sánh 2/3^2 + 2/3^4 +..+ 2/3^100 với 1/4 23/07/2021 Bởi Mackenzie so sánh 2/3^2 + 2/3^4 +..+ 2/3^100 với 1/4
Giải thích các bước giải: $\begin{array}{l}A = \dfrac{2}{{{3^2}}} + \dfrac{2}{{{3^4}}} + … + \dfrac{2}{{{3^{100}}}}\\ \Rightarrow \dfrac{1}{{{3^2}}}A = \dfrac{2}{{{3^4}}} + \dfrac{2}{{{3^6}}} + … + \dfrac{2}{{{3^{102}}}}\\ \Rightarrow A – \dfrac{1}{{{3^2}}}A = \dfrac{2}{{{3^2}}} – \dfrac{2}{{{3^{102}}}}\\ \Leftrightarrow \dfrac{8}{9}A = \dfrac{2}{9} – \dfrac{2}{{{3^{102}}}}\\ \Leftrightarrow A = \dfrac{1}{4} – \dfrac{1}{{{{4.3}^{100}}}}\\ \Rightarrow A < \dfrac{1}{4}\end{array}$ Ta có đpcm Bình luận
Đáp án: `2/3^2 + 2/3^4 +…+ 2/3^100 < 1/4` Giải thích các bước giải: Đặt `P=2/3^2+2/3^4+…+2/3^100` `=> 1/3^2P=1/3^2(2/3^2+2/3^4+…+2/3^100)` `=> 1/9P=2/3^4+2/3^6+…+2/3^102` `=> P-1/9P=(2/3^2+2/3^4+…+2/3^100)-(2/3^4+2/3^6+…+2/3^102)` `=> 8/9 P=2/3^2-2/3^102` `=> P = (2/3^2 – 2/3^102) . 9/8` `=>P=1/4-1/(3^100 .4) <1/4` Vậy `2/3^2 + 2/3^4 +…+ 2/3^100 < 1/4` Bình luận
Giải thích các bước giải:
$\begin{array}{l}
A = \dfrac{2}{{{3^2}}} + \dfrac{2}{{{3^4}}} + … + \dfrac{2}{{{3^{100}}}}\\
\Rightarrow \dfrac{1}{{{3^2}}}A = \dfrac{2}{{{3^4}}} + \dfrac{2}{{{3^6}}} + … + \dfrac{2}{{{3^{102}}}}\\
\Rightarrow A – \dfrac{1}{{{3^2}}}A = \dfrac{2}{{{3^2}}} – \dfrac{2}{{{3^{102}}}}\\
\Leftrightarrow \dfrac{8}{9}A = \dfrac{2}{9} – \dfrac{2}{{{3^{102}}}}\\
\Leftrightarrow A = \dfrac{1}{4} – \dfrac{1}{{{{4.3}^{100}}}}\\
\Rightarrow A < \dfrac{1}{4}
\end{array}$
Ta có đpcm
Đáp án:
`2/3^2 + 2/3^4 +…+ 2/3^100 < 1/4`
Giải thích các bước giải:
Đặt `P=2/3^2+2/3^4+…+2/3^100`
`=> 1/3^2P=1/3^2(2/3^2+2/3^4+…+2/3^100)`
`=> 1/9P=2/3^4+2/3^6+…+2/3^102`
`=> P-1/9P=(2/3^2+2/3^4+…+2/3^100)-(2/3^4+2/3^6+…+2/3^102)`
`=> 8/9 P=2/3^2-2/3^102`
`=> P = (2/3^2 – 2/3^102) . 9/8`
`=>P=1/4-1/(3^100 .4) <1/4`
Vậy `2/3^2 + 2/3^4 +…+ 2/3^100 < 1/4`