ND
1
K
Khách
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PK
0
NB
2
KN
26 tháng 4 2019
\(S=\frac{5}{2^2}+\frac{5}{3^2}+\frac{5}{4^2}+...+\frac{5}{100^2}\)
\(\Rightarrow S=5\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\right)\)
\(\Rightarrow S< 5\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\right)\)
\(\Rightarrow S< 5\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\right)\)
\(\Rightarrow S< 5\left(1-\frac{1}{100}\right)< 5.1=5\)
Vậy S < 5 (đpcm)
KN
26 tháng 4 2019
\(S=\frac{5}{2^2}+\frac{5}{3^2}+\frac{5}{4^2}+...+\frac{5}{100^2}\)
\(\Rightarrow S=5\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\right)\)
\(\Rightarrow S>5\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{100.101}\right)\)
\(\Rightarrow S>5\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{100}-\frac{1}{101}\right)\)
\(\Rightarrow S>5\left(\frac{1}{2}-\frac{1}{101}\right)\)
\(\Rightarrow S>5\left(\frac{101}{202}-\frac{2}{202}\right)\)
\(\Rightarrow S>5.\frac{99}{202}=\frac{495}{202}>2\)
Vậy S > 2 ( đpcm)
LT
0
NT
2
AK
11 tháng 3 2018
\(S=\frac{5}{2^2}+\frac{5}{3^2}+\frac{5}{4^2}+...+\frac{5}{100^2}\)
\(S=5.\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\right)\)
Ta có : \(\frac{1}{2^2}>\frac{1}{2.3},\frac{1}{3^2}>\frac{1}{3.4},\frac{1}{4^2}>\frac{1}{4.5},...,\frac{1}{100^2}>\frac{1}{100.101}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}< \frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{100.101}\)
\(\Rightarrow5.\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}\right)>5.\left(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{100.101}\right)\)
\(\Rightarrow S>5.\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{100}-\frac{1}{101}\right)\)
\(\Rightarrow S>5.\left(\frac{1}{2}-\frac{1}{101}\right)\)
\(\Rightarrow S>5.\frac{99}{202}\)
\(\Rightarrow S>\frac{495}{202}>\frac{404}{202}=2\)
\(\Rightarrow S>2\)
AK
11 tháng 3 2018
\(CM:S< 5\)
Ta có :
\(\frac{1}{2^2}< \frac{1}{1.2},\frac{1}{3^2}< \frac{1}{2.3},...,\frac{1}{100^2}< \frac{1}{99.100}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}< \frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{99.100}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< 1-\frac{1}{100}\)
\(\Rightarrow5.\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}\right)< 5.\frac{99}{100}\)
\(\Rightarrow S< \frac{495}{100}< \frac{500}{100}\)
\(\Rightarrow S< 5\)
LV
2
10 tháng 5 2018
\(S=\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...+\frac{1}{99^2}+\frac{1}{100^2}\)
Ta có:
\(\frac{1}{3^2}=\frac{1}{9}< \frac{1}{6}=\frac{1}{2.3}\)
\(\frac{1}{4^2}=\frac{1}{16}< \frac{1}{12}=\frac{1}{3.4}\)
Tương tự đến hết thì:
\(\frac{1}{100^2}=\frac{1}{10000}< \frac{1}{9900}=\frac{1}{99.100}\)
=> \(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...+\frac{1}{99^2}+\frac{1}{100^2}< \frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{98.99}+\frac{1}{99.100}\)
=>\(S< \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{98}-\frac{1}{99}+\frac{1}{99}-\frac{1}{100}\)
=>\(S< \frac{1}{2}-\frac{1}{100}< \frac{1}{2}\)
=> \(S< \frac{1}{2}\)
10 tháng 5 2018
nhận xét
\(\frac{1}{3^2}=\frac{1}{3.3}< \frac{1}{2\cdot3}=\frac{1}{2}-\frac{1}{3}\)
\(\frac{1}{4^2}=\frac{1}{4\cdot4}< \frac{1}{3\cdot4}=\frac{1}{3}-\frac{1}{4}\)
...........................................
\(\frac{1}{99^2}=\frac{1}{99\cdot99}< \frac{1}{98\cdot99}=\frac{1}{98}-\frac{1}{99}\)
\(\frac{1}{100^2}=\frac{1}{100\cdot100}< \frac{1}{99\cdot100}=\frac{1}{99}-\frac{1}{100}\)
ta có
S=\(\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{99^2}+\frac{1}{100^2}< \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{98}-\frac{1}{99}+\frac{1}{99}-\frac{1}{100}\)
S=\(\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{99^2}+\frac{1}{100^2}< \frac{1}{2}-\frac{1}{100}< \frac{1}{2}\)
=>S<\(\frac{1}{2}\)
Vậy S<\(\frac{1}{2}\)
DN
0
TA
0
KS
0
5/22 + 5/32 + 5/42 +...+ 5/1002 < 5/1.2 + 5/2.3 +5/3.4 +...+ 5/99.100
5/2.2 +5/3.3 + 5/4.4 +...+ 5/100.100 < 5. ( 1/1.2 + 1/2.3 +1/3.4 +..+ 1/99.100)
5/2.2 +5/3.3 + 5/4.4 +...+ 5/100.100 < 5. (1/1 -1/2 +1/2 -1/3 +1/3-1/4 +...+ 1/99-1/100)
5/2.2 +5/3.3 + 5/4.4 +...+ 5/100.100 < 5. (1/1-1/100)
5/2.2 +5/3.3 + 5/4.4 +...+ 5/100.100 < 5. ( 100/100 -1/100)
5/2.2 +5/3.3 + 5/4.4 +...+ 5/100.100 < 5. 99/100
5/2.2 +5/3.3 + 5/4.4 +...+ 5/100.100 < 99/20
mình chỉ giải tới đây thôi vì đã dễ rồi