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VD
16 tháng 4 2023
=> 4S = 1 + 2/4 + 3/4^2 +...+ 2023/4^2022
=> 4S-S = 1 + (2/4-1/4) + (3/4^2 - 2/4^2) +...+ (2023/4^2022 - 2022/4^2022) - 2023/4^2023
=> 3S = 1 + 1/4 + 1/4^2 +...+ 1/4^2022 - 2023/4^2023
=> 12S = 4 + 1 + 1/4 +... + 1/4^2021 - 2023/4^2022
=> 12S - 3S = 4 + (1-1) + (1/4-1/4) +... + (1/4^2021 - 1/4^2021) - 1/4^2022 - 2023/4^2022 + 2023/4^2023
=> 9S = 4 - 1/4^2022 - 2023/4^2022 + 2023/4^2023
= 4- 2024/4^2022 + 2023/4^2023
Do 2024/4^2022 > 2024/4^2023 > 2023/4^2023 nên - 2024/4^2022 + 2023/4^2023 < 0
=> 9S < 4 < 9/2
=> S < 1/2 (đpcm)
16 tháng 4 2023
Ta có S = \(\dfrac{1}{4}+\dfrac{2}{4^2}+\dfrac{3}{4^3}+...+\dfrac{2023}{4^{2023}}\)
4S = \(1+\dfrac{2}{4}+\dfrac{3}{4^2}+...+\dfrac{2023}{4^{2022}}\)
4S - S = ( \(1+\dfrac{2}{4}+\dfrac{3}{4^2}+...+\dfrac{2023}{4^{2022}}\) ) - ( \(\dfrac{1}{4}+\dfrac{2}{4^2}+\dfrac{3}{4^3}+...+\dfrac{2023}{4^{2023}}\))
3S = 1 + \(\dfrac{1}{4}+\dfrac{1}{4^2}+...+\dfrac{1}{4^{2022}}-\dfrac{2023}{4^{2023}}\)
Đặt A = 1 + \(\dfrac{1}{4}+\dfrac{1}{4^2}+...+\dfrac{1}{4^{2022}}\)
4A = 4 + 1 + \(\dfrac{1}{4}+...+\dfrac{1}{4^{2021}}\)
4A - A = ( 4 + 1 + \(\dfrac{1}{4}+...+\dfrac{1}{4^{2021}}\)) - ( 1 + \(\dfrac{1}{4}+\dfrac{1}{4^2}+...+\dfrac{1}{4^{2022}}\))
3A = 4 - \(\dfrac{1}{4^{2022}}\)
A = ( 4 - \(\dfrac{1}{4^{2022}}\)) : 3 = \(\dfrac{4}{3}-\dfrac{1}{4^{2022}\cdot3}\)
⇒ 3S = \(\dfrac{4}{3}-\dfrac{1}{4^{2022}\cdot3}\) - \(\dfrac{2023}{4^{2023}}\)
S = ( \(\dfrac{4}{3}-\dfrac{1}{4^{2022}\cdot3}\) - \(\dfrac{2023}{4^{2023}}\)) : 3 = \(\dfrac{4}{9}-\dfrac{1}{4^{2022}\cdot3^2}-\dfrac{1}{4^{2023}\cdot3}< \dfrac{4}{9}< \dfrac{1}{2}\)
Vậy S < \(\dfrac{1}{2}\)
M
0
23 tháng 1 2018
^ là dấu phân số nhé
cho A=1^1.2+1^2.3+...+1^2014.2015
1^1.2>1^4; 1^2.3>2^42; 1^3.4>3^43;...;1^2014.2015>2014^42014
mà A=1^1.2+1^2.3+...+1^2104.2015=1-1^2+1^2-1^3+1^3+...+1^2014-1^2015
A=1-1^2015=2014^2015
mà 2014^2015>1^2>S nên 1^2>S
NQ
1
MT
0
Lời giải:
$S=\frac{1}{4}+\frac{2}{4^2}+\frac{3}{4^3}+...+\frac{2023}{4^{2023}}$
$4S=1+\frac{2}{4}+\frac{3}{4^2}+\frac{4}{4^3}+.....+\frac{2023}{4^{2022}}$
$\Rightarrow 4S-S=1+\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+....+\frac{1}{4^{2022}}-\frac{2023}{4^{2023}}$
$\Rightarrow 3S+\frac{2023}{4^{2023}}=1+\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+....+\frac{1}{4^{2022}}$
$\Rightarrow 4(3S+\frac{2023}{4^{2023}})=4+1+\frac{1}{4}+\frac{1}{4^2}+....+\frac{1}{4^{2021}}$
$\Rightarrow 4(3S+\frac{2023}{4^{2023}})-(3S+\frac{2023}{4^{2023}})=4-\frac{1}{4^{2022}}$
$\Rightarrow 3(3S+\frac{2023}{4^{2023}})=4-\frac{1}{4^{2022}}$
$9S=4-\frac{1}{4^{2022}}-\frac{2.2023}{4^{2023}}<4$
$\Rightarrow S< \frac{4}{9}< \frac{1}{2}$