1/22+1/42+...+1/142 <1/2
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a, 27.75 + 25.27 – 150
= 2025 + 675 – 150 = 2550
b, 142 – [50 – ( 2 3 .10 – 2 3 .5)]
= 142 – [50 – (80 – 40)] = 132
c, 375:{32 – [4+( 5 . 3 2 – 42]} – 14
= 375:{32 – [4+(45 – 42)]} – 14
= 375:(32 – 7) – 14 = 15 – 14 = 1
d, {210:[16+3.(6+3. 2 2 )]} – 3
= [210:(16+3.18)] – 3
= 210 : 70 – 3 = 3 – 3 = 0
a, 27.75 + 25.27 – 150
= 2025 + 675 – 150 = 2550
b, 142 – [50 – ( 2 3 .10 – 2 3 .5)]
= 142 – [50 – (80 – 40)] = 132
c, 375:{32 – [4+( 5 . 3 2 – 42]} – 14
= 375:{32 – [4+(45 – 42)]} – 14
= 375:(32 – 7) – 14 = 15 – 14 = 1
d, {210:[16+3.(6+3. 2 2 )]} – 3
= [210:(16+3.18)] – 3
= 210 : 70 – 3 = 3 – 3 = 0
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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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\(a,\Rightarrow2^{x-1}=24-\left(16-3\right)-3\\ \Rightarrow2^{x-1}=24-13-3\\ \Rightarrow2^{x-1}=8=2^3\\ \Rightarrow x-1=3\Rightarrow x=4\\ b,\Rightarrow\left(19x+50\right):14=25-16=9\\ \Rightarrow19x+50=126\\ \Rightarrow x=4\)
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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
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\(3+2^{x-1}=24-\left[4^2-\left(2^2-1\right)\right]\)
\(\Leftrightarrow2^{x-1}=24-16+3-3\)
\(\Leftrightarrow x-1=3\)
hay x=4
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Lời giải:
Gọi vế trái là $A$
$2A=\frac{2}{2^2}+\frac{2}{4^2}+\frac{2}{6^2}+...+\frac{2}{2022^2}$
Xét số hạng tổng quát:
$\frac{2}{n^2}$. Ta sẽ cm $\frac{2}{n^2}< \frac{1}{(n-1)n}+\frac{1}{n(n+1)}(*)$
$\Leftrightarrow \frac{2}{n^2}< \frac{n+1+n-1}{n(n-1)(n+1)}$
$\Leftrightarrow \frac{2}{n^2}< \frac{2}{(n-1)(n+1)}$
$\Leftrightarrow \frac{2}{n^2}< \frac{2}{n^2-1}$ (luôn đúng)
Thay $n=2,4,...., 2022$ vào $(*)$ ta có:
$\frac{2}{2^2}< \frac{1}{1.2}+\frac{1}{2.3}$
$\frac{2}{4^2}< \frac{1}{3.4}+\frac{1}{4.5}$
.......
Suy ra: $2A< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{2022.2023}$
$2A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+....+\frac{1}{2022}-\frac{1}{2023}$
$2A< 1-\frac{1}{2023}< 1$
$\Rightarrow A< \frac{1}{2}$
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a:
Số số hạng trong dãy M là:
(1002-12):10+1=100(số)
=>Sẽ có 50 cặp (1002;992); (982;972);....;(22;12) có hiệu bằng 10
\(M=1002-992+982-972+...+22-12\)
\(=\left(1002-992\right)+\left(982-972\right)+...+\left(22-12\right)\)
\(=10+10+...+10\)
=10*50=500
b: \(N=\left(202+182+...+42+22\right)-\left(192+172+...+32+12\right)\)
\(=\left(202-192\right)+\left(182-172\right)+...+\left(22-12\right)\)
=10+10+...+10
=10*10=100
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