y
Bài 1:
Đặt \(A=1+\dfrac{3}{2^3}+\dfrac{4}{2^4}+...+\dfrac{100}{2^{100}}\)
\(\Rightarrow2A=2+\dfrac{3}{2^2}+\dfrac{4}{2^3}+...+\dfrac{100}{2^{99}}\)
\(\Rightarrow2A-A=\left(2+\dfrac{3}{2^2}+\dfrac{4}{2^3}+...+\dfrac{100}{2^{99}}\right)-\left(1+\dfrac{3}{2^3}+\dfrac{4}{2^4}+...+\dfrac{100}{2^{100}}\right)\)
\(\Rightarrow A=\left(2-1\right)+\dfrac{3}{2^2}+\left(\dfrac{1}{2^3}+\dfrac{1}{2^4}+...+\dfrac{1}{2^{100}}\right)-\dfrac{100}{2^{100}}\)
\(\Rightarrow A=1+\dfrac{3}{2^2}+\dfrac{1}{2^2}-\dfrac{1}{2^{100}}-\dfrac{100}{2^{100}}\)
\(\Rightarrow A=1+1-\dfrac{101}{2^{100}}\)
\(\Rightarrow A=2-\dfrac{101}{2^{100}}\)
Vậy \(A=2-\dfrac{101}{2^{100}}\)
Bài 2:
+) Xét \(x\ge-\dfrac{3}{4}\) ta có:
\(4x+3-x=15\)
\(\Rightarrow3x=12\)
\(\Rightarrow x=4\) ( t/m )
+) Xét \(x< \dfrac{-3}{4}\) ta có:
\(-4x-3-x=15\)
\(\Rightarrow-5x=18\)
\(\Rightarrow x=-3,6\) ( t/m )
Vậy x = 4 hoặc x = -3,6
Bài 1:
Đặt A = \(1+\dfrac{3}{2^3}+\dfrac{4}{2^4}+\dfrac{5}{2^5}+............+\dfrac{100}{2^{100}}\)
\(\Rightarrow2A=2+\dfrac{3}{2^2}+\dfrac{4}{2^3}+\dfrac{5}{2^4}+...........+\dfrac{100}{2^{99}}\)
\(\Rightarrow2A-A=1+\dfrac{3}{2^2}+\dfrac{1}{2^3}+\dfrac{1}{2^4}+.....+\dfrac{1}{2^{99}}-\dfrac{100}{2^{100}}\)
\(\Rightarrow A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+......+\dfrac{1}{2^{99}}-\dfrac{100}{2^{100}}\)
\(\Rightarrow2A=2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+......+\dfrac{1}{2^{98}}-\dfrac{100}{2^{99}}\)
\(\Rightarrow2A-A=2-\dfrac{100}{2^{99}}-\dfrac{1}{2^{99}}+\dfrac{100}{2^{100}}\)
\(\Rightarrow A=2-\dfrac{101}{2^{99}}+\dfrac{100}{2^{100}}\)
\(\Rightarrow A=\dfrac{2^{101}}{2^{100}}-\dfrac{202}{2^{100}}+\dfrac{100}{2^{100}}\)
\(\Rightarrow A=\dfrac{2^{101}-102}{2^{100}}\)
Bài 2:
|4x+3| - x = 15 (*)
TH1: |4x+3|=4x+3 khi 4x+3 \(\ge\) 0 =>4x \(\ge\) -3 => x \(\ge\) \(-\dfrac{3}{4}\)
=> (*) có dạng:
4x+3-x=15
3x+3=15
3x=12
x=4 > \(-\dfrac{3}{4}\) ( thỏa mãn)
TH2: |4x+3|=-4x-3 khi 4x+3 <0 => 4x<-3 => x<\(-\dfrac{3}{4}\)
=> (*) có dạng:
(-4x-3)-x=15
-5x-3=15
-5x=18
x=\(-\dfrac{18}{5}\)< \(-\dfrac{3}{4}\) ( thỏa mãn )
Vậy \(x=4\) và x = \(-\dfrac{18}{5}\) thỏa mãn đề bài.