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\(a,\Rightarrow2^{x-1}=24-\left(16-3\right)-3\\ \Rightarrow2^{x-1}=24-13-3\\ \Rightarrow2^{x-1}=8=2^3\\ \Rightarrow x-1=3\Rightarrow x=4\\ b,\Rightarrow\left(19x+50\right):14=25-16=9\\ \Rightarrow19x+50=126\\ \Rightarrow x=4\)
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\(3+2^{x-1}=24-\left[4^2-\left(2^2-1\right)\right]\) (sửa đề)
\(\Rightarrow3+2^{x-1}=24-\left[16-\left(4-1\right)\right]\)
\(\Rightarrow3+2^{x-1}=24-\left(16-3\right)\)
\(\Rightarrow3+2^{x-1}=24-13\)
\(\Rightarrow3+2^{x-1}=11\)
\(\Rightarrow2^{x-1}=11-3\)
\(\Rightarrow2^{x-1}=8\)
\(\Rightarrow2^{x-1}=2^3\)
\(\Rightarrow x-1=3\)
\(\Rightarrow x=3+1=4\)
Vậy \(x=4.\)
#\(Toru\)
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Giải:
A=1/22+1/32+1/42+...+1/92
Ta có:
1/22<1/1.2
1/32<1/2.3
1/42<1/3.4
...
1/92<1/8.9
⇒A<1/1.2+1/2.3+1/3.4+...+1/8.9
A<1/1-1/2+1/2-1/3+1/3-1/4+...+1/8-1/9
A<1/1-1/9
A<8/9
Ta có:
1/22>1/2.3
1/32>1/3.4
1/42>1/4.5
...
1/92>1/9.10
⇒A>1/2.3+1/3.4+1/4.5+...+1/9.10
A>1/2-1/3+1/3-1/4+1/4-1/5+...+1/9-1/10
A>1/2-1/10
A>2/5
Vậy 2/5<A<8/9 (đpcm)
Chúc bạn học tốt!
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\(B=\dfrac{1}{2^2}+\dfrac{1}{4^2}+\dfrac{1}{6^2}+...+\dfrac{1}{100^2}\)
\(B=\dfrac{1}{2.2}+\dfrac{1}{4.4}+...+\dfrac{1}{100.100}\)
\(B=\dfrac{1}{2}-\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{1}{4}+...+\dfrac{1}{100}-\dfrac{1}{100}\)
\(B=0+0+...+0\)
\(B=0\)
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Ta thấy \(\dfrac{1}{2^2}< \dfrac{1}{1.2}\)
\(\dfrac{1}{3^2}< \dfrac{1}{2.3}\)
......
\(\dfrac{1}{10^2}< \dfrac{1}{9.10}\)
hay \(D=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+....+\dfrac{1}{10^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{9.10}\)
\(D< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+....+\dfrac{1}{9}-\dfrac{1}{10}\)
\(D< 1-\dfrac{1}{10}=\dfrac{9}{10}< 1\) ( đpcm )
Ta có \(\dfrac{1}{2.2}\) < \(\dfrac{1}{1.2}\)
\(\dfrac{1}{3.3}\)<\(\dfrac{1}{2.3}\)
\(\dfrac{1}{4.4}\)<\(\dfrac{1}{3.4}\)
.........................
\(\dfrac{1}{10.10}\)<\(\dfrac{1}{9.10}\)
=>\(\dfrac{1}{2.2}+\dfrac{1}{3.3}+\dfrac{1}{4.4}+...+\dfrac{1}{10.10}\)\(< \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{9.10}\)
=> D < 1 - \(\dfrac{1}{10}\)
=>D < \(\dfrac{9}{10}\)
=> D < \(\dfrac{10}{10}\)
Vậy D < 1
Đặt C=\(\frac{1}{2^2}+\frac{1}{4^2}+...+\frac{1}{14^2}< \frac{1}{2^2}+\frac{2}{2.4}+\frac{2}{4.6}+...+\frac{2}{12.14}\)
\(\frac{1}{2^2}+\frac{1}{4^2}+..+\frac{1}{14^2}< \frac{1}{2^2}+\frac{4-2}{2.4}+\frac{6-4}{4.6}+..+\frac{14-12}{12.14}\)
\(\frac{1}{2^2}+\frac{1}{4^2}+..+\frac{1}{14^2}< \frac{1}{2^2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+..+\frac{1}{12}-\frac{1}{14}\)
\(\frac{1}{2^2}+\frac{1}{4^2}+..+\frac{1}{14^2}< \frac{1}{2^2}+(\frac{1}{2}-\frac{1}{14}):2\) => \(\frac{1}{2^2}+\frac{1}{4^2}+..+\frac{1}{14^2}< \frac{13}{28}\)
Mà 13 / 28 < 1 / 2 nên C < 1/2 => bài toán được chứng minh