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Bài 1: Tính giá trị các biểu thức:
1) \(A=\frac{2}{3}.\frac{2014}{2013}-\frac{2}{3}.\frac{1}{2013}+\frac{1}{3}\)
\(=\frac{2}{3}.\left(\frac{2014}{2013}-\frac{1}{2013}\right)+\frac{1}{3}\)
\(=\frac{2}{3}.1+\frac{1}{3}\)
= 1
\(A=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\)
\(< \frac{1}{1}+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)
\(=\frac{1}{1}+\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)
\(=\frac{1}{1}+\frac{1}{1}=2\)
\(\Rightarrow\)\(A< 2\left(đpcm\right)\)
chúc bạn học tốt!!!
Bài 6 :
2S = 6 + 3 + 3/2 + ... + 3/2^8
2S = 6 - 3/2^9 + S
S = 6 - 3/2^9
Vậy S = 6 - 3/2^9
Bài 7 :
Ta có :
A = 1/1 + 1/2^2 + 1/3^2 + ... + 1/50^2 < 1 + 1/(1x2) + 1/(2x3) + ... + 1/(49x50) = 1 + 1 - 1/50 < 1 + 1 = 2
=) A < 2
Vậy A < 2
Bài 8 :
Do A = 1 + 2/(2015^2014 - 1 ) và B = 1 + 2/(2015^2014 - 3 ) mà 2/(2015^2014 -1) < 2/(2015^2014 - 3 )
=) A < B
Vậy A < B
Bài 9:
Do 196/197 > 196/(197+198) và 197/198 > 197/(197+198)
=) A > B
Vậy A > B
1) Ta có : \(\frac{x-2}{4}=\frac{5+x}{3}\)
\(\Rightarrow\left(x-2\right).3=\left(5+x\right).4\)
\(\Rightarrow3x-6=20+4x\)
\(\Rightarrow3x=26+4x\)
\(\Rightarrow3x=26+x+3x\)
\(\Rightarrow0=26+x\)
\(\Rightarrow x=0-26\)
\(\Rightarrow x=-26\)
2) Ta có : \(A=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2012}}\)
\(\Rightarrow\frac{1}{A}=1+2+2^2+...+2^{2012}\)
\(\Rightarrow\frac{2}{A}=2+2^2+2^3+...+2^{2013}\)
\(\Rightarrow\frac{2}{A}-\frac{1}{A}=\left(2+2^2+2^3+...+2^{2013}\right)-\left(1+2+2^2+...+2^{2012}\right)\)
\(\Rightarrow\frac{1}{A}=2^{2013}+1\)
\(\Rightarrow A=\frac{1}{2^{2013}+1}\)
Mk làm mẫu câu a nha
a, Có :
2A = 1+1/2+1/2^2+.....+1/2^98
A = 2A - A = (1+1/2+1/2^2+.....+1/2^98)-(1/2+1/2^2+......+1/2^99) = 1 - 1/2^99
Tk mk nha
a. \(A=\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2014}}\)
\(\Rightarrow3A=1+\frac{1}{3}+\frac{1}{3^2}+....+\frac{1}{3^{2013}}\)
\(\Rightarrow3A-A=1-\frac{1}{3^{2014}}\)
\(\Rightarrow2A=1-\frac{1}{3^{2014}}\)
\(\Rightarrow A=\left(1-\frac{1}{3^{2014}}\right):2=\frac{1}{2}-\frac{1}{3^{2014}.2}=\frac{3^{2014}-1}{3^{2014}.2}\)
b.\(B=\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^{2014}}\)
\(\Rightarrow2B=1+\frac{1}{2^2}+....+\frac{1}{2^{2013}}\)
\(\Rightarrow2B-B=1-\frac{1}{2^{2014}}\)
\(\Rightarrow B=1-\frac{1}{2^{2014}}\)