Cho M = \(\frac{4}{3}\) + \(\frac{10}{9}\) + \(\frac{28}{27}\) + ... + \(\frac{3^{98}+1}{3^{98}}\)
Chứng minh M < 100
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30 tháng 4 2017
= \(1+\frac{1}{3}+1+\frac{1}{9}+1+\frac{1}{27}+...+1+\frac{1}{3^{98}}\)\(\frac{1}{3^{98}}\)
\(=1.98+\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\right)\)
Đặt A=\(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\)
\(\Rightarrow3A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{97}}\)
\(\Rightarrow3A-A=2A=1-\frac{1}{3^{98}}\Rightarrow A=\frac{1-\frac{1}{2^{98}}}{2}< 1\)
\(\Rightarrow B=98+A< 98+1< 99< 100\)
\(\Rightarrow B< 100\)
23 tháng 4 2018
Xét \(B=\frac{4}{3}+\frac{10}{9}+...+\frac{3^{98}+1}{3^{98}}\)
\(\Leftrightarrow B=\frac{3+1}{3}+\frac{9+1}{9}+...+\frac{3^{98}+1}{3^{98}}\)
\(\Leftrightarrow B=\left(1+\frac{1}{3}\right)+\left(1+\frac{1}{9}\right)+...+\left(1+\frac{1}{3^{98}}\right)\)(có 98 cặp số hạng)
\(\Leftrightarrow B=\left(1+1+...+1\right)+\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{98}}\right)\)(có 98 số hạng 1)
\(\Leftrightarrow B=98+\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{98}}\right)\)
Đặt \(A=\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{97}}+\frac{1}{3^{98}}\)
\(\Rightarrow3A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{97}}\)
Lấy 3A-A, ta được:
\(2A=1-\frac{1}{3^{98}}\)
\(\Rightarrow A=\frac{1}{2}-\frac{1}{2\cdot3^{98}}\)(*)
Thay (*) vào biểu thức B, ta được
\(B=98+\frac{1}{2}-\frac{1}{2\cdot3^{98}}< 100\)
VẬY, B<100 (ĐPCM)
23 tháng 4 2018
Ta có :
\(B=\frac{4}{3}+\frac{10}{9}+\frac{28}{27}+...+\frac{3^{98}+1}{3^{98}}\)
\(B=\frac{3+1}{3}+\frac{9+1}{9}+\frac{27+1}{27}+...+\frac{3^{98}+1}{3^{98}}\)
\(B=\frac{3}{3}+\frac{1}{3}+\frac{9}{9}+\frac{1}{9}+\frac{27}{27}+\frac{1}{27}+...+\frac{3^{98}}{3^{98}}+\frac{1}{3^{98}}\)
\(B=1+\frac{1}{3}+1+\frac{1}{9}+1+\frac{1}{27}+...+1+\frac{1}{3^{98}}\)
\(B=\left(1+1+1+...+1\right)+\left(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{3^{98}}\right)\)
\(B=\left(1+1+1+...+1\right)+\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\right)\)
Do từ \(1\) đến \(98\) có \(98-1+1=98\) số hạng nên có \(98\) số \(1\) suy ra :
\(B=98+\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\right)\)
Đặt \(A=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\) ta có :
\(3A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{97}}\)
\(3A-A=\left(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{97}}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\right)\)
\(2A=1-\frac{1}{3^{98}}< 1\)
Mà \(2A< 1\)\(\Rightarrow\)\(A< 1\)
Do đó :
\(B=98+\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\right)< 98+1=99< 100\)
\(\Rightarrow\)\(B< 100\) ( đpcm )
Vậy \(B< 100\)
Chúc bạn học tốt ~
10 tháng 5 2016
B = \(\frac{4}{3^1}+\frac{10}{3^2}+\frac{28}{3^3}+...+\frac{3^{98}+1}{3^{98}}\)
B = \(\left(1-\frac{1}{3}\right)+\left(1-\frac{1}{3^2}\right)+\left(1-\frac{1}{3^3}\right)+...+\left(1-\frac{1}{3^{98}}\right)\)
B = \(\left(1+1+1+...+1\right)-\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\right)\)
B = \(98-\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\right)\)
=> B < 98 < 100
vậy B < 100
TN
11 tháng 5 2016
\(B=\left(1-\frac{1}{3}\right)+\left(1-\frac{1}{3^2}\right)+...+\left(1-\frac{1}{3^{98}}\right)\)
\(B=\left(1+1+1+...+1\right)-\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{98}}\right)\)
\(B=98-\left(\frac{1}{3}-\frac{1}{3^2}+...+\frac{1}{3^{98}}\right)<98\)
=>B<98<100
=>B<100 (đpcm)
MT
19 tháng 10 2015
\(B=\frac{4}{3}+\frac{10}{9}+\frac{28}{27}+...+\frac{3^{98}+1}{3^{98}}\)
\(=\frac{3+1}{3}+\frac{3^2+1}{3^2}+\frac{3^3+1}{3^3}+...+\frac{3^{98}+1}{3^{98}}\)
\(=1+\frac{1}{3}+1+\frac{1}{3^2}+1+\frac{1}{3^3}+...+1+\frac{1}{3^{98}}\)
\(=98+\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\right)\)
\(\text{Đặt }A=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\)
\(\text{rút gon cái A thì dc: }A=\frac{1}{3^{98}}-1\Rightarrow B=98+\frac{1}{3^{98}}-1=97+\frac{1}{3^{98}}\)
\(
9 tháng 5 2016
\(M=1+\frac{1}{3}+1+\frac{1}{9}+1+\frac{1}{27}+...+1+\frac{1}{3^{98}}\)
\(=1.98+\left(\frac{1}{3^1}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{98}}\right)\)
Đặt A=\(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+....+\frac{1}{3^{98}}\)
=>\(3A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{97}}\)
=>3A-A=2A=\(1-\frac{1}{3^{98}}\Rightarrow A=\frac{1-\frac{1}{3^{98}}}{2}< 1\)
=>M=98+A<98+1<99<100
=>đpcm