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b ) \(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{99.100}\)
= 1 - 1/2 + 1/2 - 1/3 + ... + 1/99 - 1/100
= 1 - 1/100
= 99/100
c ) Đặt A = \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}\)
=> A < \(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{99.100}\)
=> A < 1 - 1/2 + 1/2 - 1/3 + ... + 1/99 - 1/100= 1 - 1/100 = 99/100 < 1
Vậy \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}\)< 1
b, \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{98.99}+\)\(\frac{1}{99.100}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{98}-\frac{1}{99}+\frac{1}{99}-\frac{1}{100}\)
\(=1-\frac{1}{100}\)
\(=\frac{99}{100}\)
c,Ta thấy
\(\frac{1}{2^2}< \frac{1}{1.2}\)
\(\frac{1}{3^2}< \frac{1}{2.3}\)
\(\frac{1}{4^2}< \frac{1}{3.4}\)
\(.....\)
\(\frac{1}{100^2}< \frac{1}{99.100}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)\(< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(=1-\frac{1}{100}< 1\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< 1\left(đpcm\right)\)
a)17.85-15.17-120
=17.(85-17)-120
=17.68-12
=1156-120
=1036
b)100:[30-(6-1)^ 2]
=100 :[30-5^ 2]
= 100:[30-25]
=100:5=20
c)/-10/+4+3.2^3+(-14)
=10+4+3.8 +(-14)
=14+24+(-14)
=[14 +(-14)]+24
=0+24=24
d)20-[30-(5-1)^ 2: 2]
=20-[30-4^2:2]
=20-[30-8]
=20-22=-2
e)80-(4.5^2-3.2^3)
= 80-(4.25-3.8)
=80-(4.25-3.2.4)
=80-4.(25-3.2)
= 80-4.19
= 80-76=4
g)35-{12-[-14+6-2)]}
=35 -{12-14-6+2}
=35-12+14+6-2
=23+14+6-2
=37+6-2
=43-2=41
làm được câu a thui
\(\frac{1}{100}=\frac{1}{10.10}\);\(\frac{1}{90}=\frac{1}{9.10}\);...
Suy ra \(\frac{1}{10.10}-\frac{1}{9.10}-\frac{1}{8.9}-\frac{1}{7.8}-\frac{1}{6.7}-...-\frac{1}{1.2}\)
\(\frac{1}{10}-\frac{1}{10}-\frac{1}{10}-\frac{1}{9}-...-1-\frac{1}{2}\)
\(\frac{1}{10}-\frac{1}{2}\)
\(-\frac{4}{10}\)
a ) ( - 14 ) + 12 + ( - 12 ) . ( 23 - 7 )
= - 2 + ( - 12 ) . ( 8 - 7 )
= - 2 + ( - 12 ) . 1
= - 2 + ( - 12 )
= - 14
a/ ta có : 5.(27-17)2 - 611 : 69 = 5.102 - 611 :69
= 5.100 - 611-9 = 500 - 36 = 464
b/ ta có : (-15)+(-17) = -15 - 17 = -32
c/ ta có : 21.42 + 21.59 +21.52 = 21.(42 +59 +52 )
= 21. (16 + 59 + 25) = 21.100=2100
d/ ta có : 75 - (3.52 - 4.23 ) +20150 = 75 - (3.25 - 4.8) +1
= 75 - 43 +1 = 33
\(a,721+100:\left[9-2.\left(4^2-14\right)^2\right]\)
\(=721+100:\left[9-2.\left(2\right)^2\right]\)
\(=721+100:\left[9-2.4\right]\)
\(=721+100:\left[9-8\right]\)
\(=721+100:1\)
\(=721+100\)
\(=821\)
\(b,\left|-15\right|.\left(-42\right).\left|21\right|\)
\(=15.\left(-42\right).21\)
\(=-630.21\)
\(=-13230\)