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A = \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)
A < \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)
A < \(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\)
A < 1 - \(\frac{1.}{100}\)
A < \(\frac{99}{100}< \frac{199}{100}\)
=> A < \(\frac{199}{100}\)
b,
S = \(\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}...\frac{99}{10^2}\)
S = \(\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}...\frac{9.11}{10.10}\)
S = \(\frac{1.3.2.4.3.5.4.6.5.7...9.11}{2.2.3.3.4.4...10.10}\)
S = \(\frac{1.2.3^2.4^2.5^2...9^2.10.11}{2^2.3^3.4^2...10^2}\)
S = \(\frac{1.11}{2.10}\)
S = \(\frac{11}{20}\)
\(\left(\frac{1}{2}-1\right)\left(\frac{1}{3}-1\right)\left(\frac{1}{4}-1\right)...\left(\frac{1}{999}-1\right)=\frac{-1}{2}.\frac{-2}{3}.\frac{-3}{4}...\frac{-998}{999}=\frac{1}{999}\)
-------
\(\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}...\frac{99}{10^2}=\frac{3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}...\frac{9.11}{10.10}=\frac{\left(2.3.4...9\right).\left(3.4.5...11\right)}{\left(2.3.4...10\right).\left(2.3.4...10\right)}=\frac{1.11}{10.2}=\frac{11}{20}\)
Lời giải:
Ta có:
\(\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}....\frac{99}{10^2}=\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}...\frac{9.11}{10^2}\)
\(=\frac{(1.3).(2.4).(3.5)...(9.11)}{2^2.3^2.4^2...10^2}\)
\(=\frac{(1.2.3...9)(3.4.5...11)}{(2.3.4...10)(2.3.4..10)}\)
\(=\frac{1}{10}.\frac{11}{2}=\frac{11}{20}\)
biết làm bài 1 thôi
\(\left(\frac{1}{2}+1\right)\times\left(\frac{1}{3}+1\right)\times\cdot\cdot\cdot\times\left(\frac{1}{999}+1\right)\)
= \(\frac{3}{2}\times\frac{4}{3}\times\frac{5}{4}\times\cdot\cdot\cdot\times\frac{1000}{999}\)
lượt bỏ đi còn :
\(\frac{1000}{2}=500\)
Bài 2:
B = (1/2 - 1)(1/3 -1).(1/4 -1)...(1/1000 - 1)
B = (1/2 - 2/2).(1/3 - 3/3)...(1/1000 - 1000/1000)
B = (-1/2).(-2/3)...(-999/1000)
Xét dãy số: 1; 2 ;3;...999
Dãy số trên có 999 số hạng vậy B là tích của 999 số âm
B = - 1/1000
Bài 1:
1/6 + -5/7 : x = - 7/18
5/7: x = 1/6 + 7/18
5/7: x = 3/18 + 7/18
5/7: x = 5/9
x = 5/7 : 5/9
x = 5/7 x 9/5
x = 9/7
Vậy x = 9/7
Bài 2:
Câu a:
(1/2 + 1).(1/3 + 1).(1/4 + 1)...(1/999 + 1)
= (1/2 + 2/2).(1/3 + 3/3)...(1/999 + 999/999)
= 3/2.4/3....1000/999
= 1000/2
= 500
a,\(\frac{31}{1000}\)
b,\(\frac{8}{108}\)
c,0
a,\(\frac{49}{97}\)
b,\(\frac{-1}{4751}\)