Chứng minh :
1,C=\(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{70}.C< \frac{3}{4}\)
2,D=\(\frac{1}{5^2}+\frac{1}{9^2}+...+\frac{1}{409^2}< \frac{1}{12}\)
3,E=\(\frac{5}{5.8.11}+\frac{5}{8.11.14}+...+\frac{5}{302.305.308}< \frac{1}{48}\)
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11 tháng 12 2019
a) \(\frac{15}{12}+\frac{5}{13}-\frac{3}{12}-\frac{18}{13}=\left(\frac{15}{12}-\frac{3}{12}\right)+\left(\frac{5}{13}-\frac{18}{13}\right)\)
\(=1+\left(-1\right)\)
\(=0\)
b) \(\frac{11}{24}-\frac{5}{41}+\frac{13}{24}+0,5-\frac{36}{41}=\left(\frac{11}{24}+\frac{13}{24}\right)+\left(-\frac{5}{41}-\frac{36}{41}\right)+0,5\)
\(=1+\left(-1\right)+0,5\)
\(=0,5\)
_Học tốt nha_
11 tháng 12 2019
a, \(\frac{15}{12}\)+ \(\frac{5}{13}\)- \(\frac{3}{12}\)-\(\frac{18}{13}\)
= \(\frac{5}{4}\)+ \(\frac{5}{13}\) - \(\frac{1}{4}\) - \(\frac{18}{13}\)
= \(\left(\frac{5}{4}-\frac{1}{4}\right)\)+ \(\left(\frac{5}{13}-\frac{18}{13}\right)\)
= 1 - 1 = 0
b, \(\frac{11}{24}\)- \(\frac{5}{41}\)+ \(\frac{13}{24}\)+ 0,5 - \(\frac{36}{41}\)
= \(\left(\frac{11}{24}+\frac{13}{24}\right)\)- \(\left(\frac{5}{41}+\frac{36}{41}\right)\)+ 0,5
= 1 - 1 + 0,5 = 0,5
c, \(\left(-\frac{3}{4}+\frac{2}{3}\right):\frac{5}{11}+\left(-\frac{1}{4}+\frac{1}{3}\right):\frac{5}{11}\)
=\(\left(-\frac{3}{4}+\frac{2}{3}\right).\frac{11}{5}+\left(-\frac{1}{4}+\frac{1}{3}\right).\frac{5}{11}\)
= \(\frac{11}{5}.\left(-\frac{3}{4}+\frac{2}{3}-\frac{1}{4}+\frac{1}{3}\right)\)
= \(\frac{11}{5}.\left[\left(-\frac{3}{4}-\frac{1}{4}\right)+\left(\frac{2}{3}+\frac{1}{3}\right)\right]\)
= \(\frac{11}{5}.\left[\left(-1\right)+1\right]\)
= 0
d, \(\left(-3\right)^2.\left(\frac{3}{4}-0,25\right)-\left(3\frac{1}{2}-1\frac{1}{2}\right)\)
= \(9.\left(0,75-0,25\right)-2\)
= 9. 0,5 - 2 = 2,5
e, \(\frac{13}{25}+\frac{6}{41}-\frac{38}{25}+\frac{35}{41}-\frac{1}{2}\)
= \(\left(\frac{13}{25}-\frac{38}{25}\right)+\left(\frac{6}{41}+\frac{35}{41}\right)-\frac{1}{2}\)
= -1 + 1 - \(\frac{1}{2}\)
= \(-\frac{1}{2}\)
9 tháng 4 2018
a) 2/7+-3/8+11/7+1/3+1/7+5/-8
=(2/7+11/7+1/7)+(3/8+-5/8)+1/3
=2+2+1/3
=4+1/3
=13/3
b) -3/8+12/25+5/-8+2/-5+13/25
=(-3/8+-5/8)+(12/25+13/25)+-2/5
=-1+1+-2/5
=0+-2/5
=-2/5
c)7/8+1/8*3/8+1/8*5/8
=7/8+1/8*(3/8+5/8)
=7/8+1/8*1
=7/8+1/8
=1
9 tháng 4 2018
a) 2/7+-3/8+11/7+1/3+1/7+5/-8
=(2/7+11/7+1/7)+(3/8+-5/8)+1/3
=2+2+1/3
=4+1/3
=13/3
b) -3/8+12/25+5/-8+2/-5+13/25
=(-3/8+-5/8)+(12/25+13/25)+-2/5
=-1+1+-2/5
=0+-2/5
=-2/5
c)7/8+1/8*3/8+1/8*5/8
=7/8+1/8*(3/8+5/8)
=7/8+1/8*1
=7/8+1/8
=1
LH
22 tháng 8 2019
a, \(\frac{1}{4}+\frac{5}{12}-\frac{1}{13}-\frac{7}{8}\)
\(=\left(\frac{1}{4}+\frac{5}{12}\right)-\left(\frac{1}{13}+\frac{7}{8}\right)\)
\(=\frac{2}{3}-\frac{99}{104}\)
\(=-\frac{89}{312}\)
b, \(11\frac{3}{13}-2\frac{4}{7}+5\frac{3}{13}\)
\(=\left(11\frac{3}{13}+5\frac{3}{13}\right)-2\frac{4}{7}\)
\(=\frac{214}{13}-\frac{18}{7}\)
\(=\frac{1264}{91}\)
c, \(\left(6\frac{4}{9}+3\frac{7}{11}\right)-4\frac{4}{9}\)
\(=6\frac{4}{9}+3\frac{7}{11}-4\frac{4}{9}\)
\(=\left(6\frac{4}{9}-4\frac{4}{9}\right)+3\frac{7}{11}\)
\(=2+3\frac{7}{11}\)
\(=5\frac{7}{11}\)
\(=\frac{62}{11}\)
d, \(\left(6,17+3\frac{5}{9}-2\frac{36}{97}\right)\left(\frac{1}{3}-0,25-\frac{1}{12}\right)\)
\(=\left(6,17+3\frac{5}{9}-2\frac{36}{97}\right)\left(\frac{1}{3}-\frac{1}{4}-\frac{1}{12}\right)\)
\(=\left(6,17+3\frac{5}{9}-2\frac{36}{97}\right)\cdot0\)
\(=0\)
e, \(-1,5\cdot\left(1+\frac{2}{3}\right)\)
\(=-\frac{3}{2}\cdot\frac{5}{3}\)
\(=-\frac{5}{2}\)
f, Đặt \(A=1^2+2^2+3^2+...+100^2\)
\(=1+2\left(3-1\right)+3\left(4-1\right)+...+100\left(101-1\right)\)
\(=1+2\cdot3-2+3\cdot4-3+...+100\cdot101-100\)
\(=\left(2\cdot3+3\cdot4+...+100\cdot101\right)-\left(1+2+3+...+100\right)\)
Đặt B = 2 . 3 + 3 . 4 + ... + 100 . 101
3B = 2 . 3 ( 4 - 1 ) + 3 . 4 ( 5 - 2 ) + ... + 100 . 101 . ( 102 - 99 )
3B = 2 . 3 . 4 - 1 . 2 . 3 + 3 . 4 . 5 - 2 . 3 . 4 + ... + 100 . 101 . 102 - 99 . 100 . 101
3B = 100 . 101 . 102
B = \(\frac{100\cdot101\cdot102}{3}\)
B = 343400
Thay B vào A. Ta được :
\(A=343400-\left(1+2+3+...+100\right)\)
Thay C = 1 + 2 + 3 + ... + 100
Dãy số 1; 2; 3; ...; 100 có số số hạng là:
( 100 - 1 ) : 1 + 1 = 100 ( số hạng )
Tổng của dãy số đó là :
( 100 + 1 ) . 100 : 2 = 5050
=> C = 5050
Thay C vào A. Ta được :
\(A=343400-5050\)
\(A=338350\)
Vậy A = 338350
5 tháng 9 2019
A = 5/7.(1+9/13) − 5/7.9/13
A= 5/7.(1+9/13 - 9/13)
A = 5/7.1
A = 5/7
B = 11/24 − 5/41 + 13/24 + 0.5 − 36/41
B = (11/24 + 13/24) - (5/41 + 36/41) + 0.5
B = 1 - 1 + 0.5
B = 0.5
C = −4/13.5/17 + (−12/13).4/17 + 4/13
C = 4/13.(-5/17) + (−12/13).4/17 + 4/13
C = 4/13.(-5/17 + 1) + (−12/13).4/17
C = 4/13.(−12/17) + (−12/13).4/17
C = (4.-12)/(13.17) + (−12/13).4/17
C = 4/17.(−12/13) + (−12/13).4/17
C = 4/17.(−12/13).2
C = 96/221
D = (4/3 − 3/2)2 − 2.∣−1/9∣ + (−5/18)
D = (4/3 − 3/2)2 − 2.1/9+ (−5/18)
D = -1/62 - 2/9+ (−5/18)
D = -1/12 - ( 2/9+ (−5/18) )
D = -1/12 - ( 4/18+ (−5/18) )
D = -1/12 - (-1/18)
D = -1/12 + 1/18
D = -3/36 + 2/36
D = -1/36
E = (−3/4 + 2/3):5/11 + (−1/4 + 1/3):5/11
E = (−3/4 + 2/3 + (−1/4) + 1/3):5/11
E = ((−3/4 + (−1/4)) + (2/3 + + 1/3)):5/11
E = ( - 1 + 1):5/11
E = 0:5/11
E = 0
