\(\dfrac{9}{9.19}+\dfrac{9}{19.29}+...+\dfrac{1}{89.99}\)
tính tổng
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Tính nhanh:
\(\dfrac{1}{19}+\dfrac{9}{19.29}+\dfrac{9}{29.39}+...+\dfrac{9}{1999.2009}\)
\(\dfrac{1}{19}+\dfrac{9}{19\cdot29}+...+\dfrac{9}{1999\cdot2009}\)
\(=\dfrac{1}{19}+\dfrac{9}{10}\left(\dfrac{10}{19\cdot29}+...+\dfrac{10}{1999\cdot2009}\right)\)
\(=\dfrac{1}{19}+\dfrac{9}{10}\left(\dfrac{1}{19}-\dfrac{1}{2009}\right)\)
\(=\dfrac{1}{19}+\dfrac{1791}{38171}=\dfrac{200}{2009}\)
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Tính nhanh:
\(B=\dfrac{1}{19}+\dfrac{9}{19.29}+\dfrac{9}{29.39}+...+\dfrac{9}{1999.2009}\)
\(A=\dfrac{1}{19}+\left(\dfrac{9}{19\cdot29}+\dfrac{9}{29\cdot39}+...+\dfrac{9}{1999\cdot2009}\right)\)
\(A=\dfrac{1}{19}+\dfrac{9}{10}\left(\dfrac{10}{19\cdot29}+\dfrac{10}{29\cdot39}+...+\dfrac{10}{1999\cdot2009}\right)\)
\(A=\dfrac{1}{19}+\dfrac{9}{10}\left(\dfrac{1}{19}-\dfrac{1}{29}+\dfrac{1}{29}-\dfrac{1}{39}+...+\dfrac{1}{1999}-\dfrac{1}{2009}\right)\)
\(A=\dfrac{1}{19}+\dfrac{9}{10}\left(\dfrac{1}{19}-\dfrac{1}{2009}\right)\)
\(A=\dfrac{1}{19}+\dfrac{9}{10}\cdot\dfrac{1990}{38171}\)
\(A=\dfrac{1}{19}+\dfrac{1791}{38171}\)
\(A=\dfrac{200}{2009}\)
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B=1/19+(9/19.29+9/29.39+...+9/1999.2009)
B=1/19+9/10+(10/19.29+10/29.39+.....+10/1999.2009
B=1/19+9/10+(1/19-1/29+1/29-1/39+....+1/1999-1/2009)
B=1/19+9/10+(1/19-1/2009)
B=1/19+9/10.1990/38171
B=1/19+1791/38171
B=200/2009
Vậy B= 200/2009
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(\(\dfrac{1}{4\cdot9}+\dfrac{1}{9\cdot14}+\dfrac{1}{14\cdot19}+...+\dfrac{1}{44\cdot49}\)) .Tính tổng
=(54⋅9+59⋅14+514⋅19+...+544⋅49)⋅1−(3+5+7+...+49)89=(54⋅9+59⋅14+514⋅19+...+544⋅49)⋅1−(3+5+7+...+49)89=15⋅(14−19+19−114+114−119+...+144−149)⋅(1−(52⋅24)289)=15⋅(14−19+19−114+114−119+...+144−149)⋅(1−(52⋅24)289)=15⋅(14−149)⋅1−62489=15⋅(14−149)⋅1−62489=15⋅45196⋅−62389=15⋅45196⋅−62389=−928
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\(\dfrac{1}{4\cdot9}+\dfrac{1}{9\cdot14}+\dfrac{1}{14\cdot19}+...+\dfrac{1}{44\cdot49}\)
\(=\dfrac{1}{5}\left(\dfrac{9-4}{4\cdot9}+\dfrac{14-9}{9\cdot14}+\dfrac{19-14}{14\cdot19}+...+\dfrac{49-44}{44\cdot49}\right)\)
\(=\dfrac{1}{5}\cdot\left(\dfrac{1}{4}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{14}+\dfrac{1}{14}-\dfrac{1}{19}+....+\dfrac{1}{44}-\dfrac{1}{49}\right)\)
\(=\dfrac{1}{5}\cdot\left(\dfrac{1}{4}-\dfrac{1}{49}\right)\)
\(=\dfrac{1}{5}\cdot\dfrac{45}{196}\)
\(=\dfrac{9}{196}\)
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Câu 3. (2 điểm) Tính nhanh tổng sau
S = 1 + \(\dfrac{1}{3}\) + \(\dfrac{1}{9}\) + \(\dfrac{1}{27}\) +\(\dfrac{1}{81}\) + \(\dfrac{1}{243}\)+ \(\dfrac{1}{729}\)
S= 1/1 + 1/3 + 1/9 + 1/27 + 1/81 + 1/243 + 1/729
S= 3 x ( 1/1 + 1/3 + 1/9 + 1/27 + 1/81 + 1/243 + 1/729 )
S = 3 +1 + 1/9 + 1/27 + 1/81 + 1/243 + 1/729
S= 3 +1 + 1/9 + 1/27 + 1/81 + 1/243 + 1/729 - 1 - 1/9 -1/27 - 1/81 - 1/243 - 1/729
S = 3 - 1/729
S= 142/729
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Cho A = \(\dfrac{2}{1.5}+\dfrac{3}{5.11}+\dfrac{4}{11.19}+\dfrac{5}{19.29}+\dfrac{6}{29.41}\)
B = \(\dfrac{1}{1.4}+\dfrac{2}{4.10}+\dfrac{3}{10.19}+\dfrac{4}{19.31}\)
Chứng tỏ rằng A > B
Lời giải:
\(2A=\frac{4}{1.5}+\frac{6}{5.11}+\frac{8}{11.19}+\frac{10}{19.29}+\frac{12}{29.41}\)
\(=1-\frac{1}{5}+\frac{1}{5}-\frac{1}{11}+\frac{1}{11}-\frac{1}{19}+...+\frac{1}{29}-\frac{1}{41}=1-\frac{1}{41}=\frac{40}{41}\)
\(\Rightarrow A=\frac{20}{21}\)
\(3B=\frac{3}{1.4}+\frac{6}{4.10}+\frac{9}{10.19}+\frac{12}{19.31}=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{10}+\frac{1}{10}-\frac{1}{19}+\frac{1}{19}-\frac{1}{31}\)
\(=1-\frac{1}{31}=\frac{30}{31}\)
\(\Rightarrow B=\frac{10}{31}=\frac{20}{62}<\frac{20}{41}\)
Do đó $A>B$
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Cho A = \(\dfrac{2}{1.5}+\dfrac{3}{5.11}+\dfrac{4}{11.19}+\dfrac{5}{19.29}+\dfrac{6}{29.41}\)
B = \(\dfrac{1}{1.4}+\dfrac{2}{4.10}+\dfrac{3}{10.19}+\dfrac{4}{19.31}\)
Chứng tỏ rằng A > B.
A.2=4/1.5+6/5.11+...+12/29.41
A.2=1-1/5+1/5-1/11+...+1/29-1/41
A.2=1-1/41
A.2=40/41
A=20/41
B.3=3/1.4+6/4.10+...+12/29.31
B.3=1-1/4+1/4-1/10+...+1/29-1/31
B.3=1-1/31
B.3=30/31
B=10/31
Vì 20/41.10/31 nên A>B
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\(A=\dfrac{2}{1.5}+\dfrac{3}{5.11}+\dfrac{4}{11.19}+\dfrac{5}{19.29}+\dfrac{6}{29.41}\)
\(\Rightarrow2A=\dfrac{4}{1.5}+\dfrac{6}{5.11}+\dfrac{8}{11.19}+\dfrac{10}{19.29}+\dfrac{12}{29.41}\)
\(\Rightarrow2A=1-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{19}+\dfrac{1}{19}-\dfrac{1}{29}+\dfrac{1}{29}-\dfrac{1}{41}\)
\(\Rightarrow2A=1-\dfrac{1}{41}=\dfrac{40}{41}\)
\(\Rightarrow A=\dfrac{40}{41}:2=\dfrac{20}{41}\)(1)
\(B=\dfrac{1}{1.4}+\dfrac{2}{4.10}+\dfrac{3}{10.19}+\dfrac{4}{19.31}\)
\(\Rightarrow3B=\dfrac{3}{1.4}+\dfrac{6}{4.10}+\dfrac{9}{10.19}+\dfrac{12}{19.31}\)
\(\Rightarrow3B=\dfrac{1}{1}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{19}+\dfrac{1}{19}-\dfrac{1}{31}\)
\(\Rightarrow3B=\dfrac{1}{1}-\dfrac{1}{31}=\dfrac{30}{31}\)
\(\Rightarrow B=\dfrac{30}{31}:3=\dfrac{10}{31}\)
\(\Rightarrow B=\dfrac{2}{2}.\dfrac{10}{31}=\dfrac{20}{62}\)
+)Ta có:\(\dfrac{20}{62}< \dfrac{20}{41}\Rightarrow B< A\)
Hay A>B(ĐPCM)
Chúc bn học tốt
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Giải:
\(A=\dfrac{2}{1.5}+\dfrac{3}{5.11}+\dfrac{4}{11.19}+\dfrac{5}{19.29}+\dfrac{6}{29.41}\)
\(2A=\dfrac{4}{1.5}+\dfrac{6}{5.11}+\dfrac{8}{11.19}+\dfrac{10}{19.29}+\dfrac{12}{29.41}\)
\(2A=\dfrac{1}{1}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{19}+\dfrac{1}{19}-\dfrac{1}{29}+\dfrac{1}{29}-\dfrac{1}{41}\)
\(2A=\dfrac{1}{1}-\dfrac{1}{41}\)
\(2A=\dfrac{40}{41}\)
\(A=\dfrac{40}{41}:2\)
\(A=\dfrac{20}{41}\)
\(B=\dfrac{1}{1.4}+\dfrac{2}{4.10}+\dfrac{3}{10.19}+\dfrac{4}{19.31}\)
\(3B=\dfrac{3}{1.4}+\dfrac{6}{4.10}+\dfrac{9}{10.19}+\dfrac{12}{19.31}\)
\(3B=\dfrac{1}{1}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{19}+\dfrac{1}{19}-\dfrac{1}{31}\)
\(3B=\dfrac{1}{10}-\dfrac{1}{31}\)
\(3B=\dfrac{21}{310}\)
\(B=\dfrac{21}{310}:3\)
\(B=\dfrac{7}{310}\)
Vì \(\dfrac{20}{41}>\dfrac{7}{310}\) nên A>B
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Hãy tính các tổng sau:
a)dfrac{1}{1cdot3}+dfrac{1}{3cdot5}+dfrac{1}{5cdot7}+dfrac{1}{7cdot9}+dfrac{1}{9cdot11}
b)dfrac{1}{4cdot7}+dfrac{1}{7cdot10}+dfrac{1}{10cdot13}+dfrac{1}{13cdot16}
c)dfrac{1}{2cdot7}+dfrac{1}{7cdot12}+dfrac{1}{12cdot17}+...
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Hãy tính các tổng sau:
a)\(\dfrac{1}{1\cdot3}\)+\(\dfrac{1}{3\cdot5}\)+\(\dfrac{1}{5\cdot7}\)+\(\dfrac{1}{7\cdot9}\)+\(\dfrac{1}{9\cdot11}\)=
b)\(\dfrac{1}{4\cdot7}\)+\(\dfrac{1}{7\cdot10}\)+\(\dfrac{1}{10\cdot13}\)+\(\dfrac{1}{13\cdot16}\)=
c)\(\dfrac{1}{2\cdot7}\)+\(\dfrac{1}{7\cdot12}\)+\(\dfrac{1}{12\cdot17}\)+...=
1100444-88888=
a)\(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\frac{1}{9.11}\)
\(=\frac{1}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+\frac{2}{9.11}\right)\)
\(=\frac{1}{2}.\left(\frac{3-1}{1.3}+\frac{5-3}{3.5}+\frac{7-5}{5.7}+\frac{9-7}{7.9}+\frac{11-9}{9.11}\right)\)
\(=\frac{1}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}\right)\)
\(=\frac{1}{2}.\left(1-\frac{1}{11}\right)\)
\(\frac{10}{22}\)
Tính (Tính hợp lí nếu có thể)a) dfrac{-7}{12}-dfrac{3}{36}b) (4-dfrac{5}{12}):2+dfrac{5}{24}c) dfrac{8}{9}+dfrac{1}{9}.dfrac{2}{13}+dfrac{1}{9}.dfrac{11}{13}d) dfrac{3}{4}.dfrac{8}{9}.dfrac{15}{16}. ... .dfrac{9999}{10000}e) dfrac{3}{1.4}+dfrac{3}{4.7}+dfrac{3}{7.10}+...+dfrac{3}{97.100}*Lưu ý: Mong các anh chị trình bày chi tiết để em có thể hiểu bài, em xin các anh chị đừng viết mỗi kết quả xong em chả biết một cái gì ;-;
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Tính (Tính hợp lí nếu có thể)
a) \(\dfrac{-7}{12}\)-\(\dfrac{3}{36}\)
b) (4-\(\dfrac{5}{12}\)):2+\(\dfrac{5}{24}\)
c) \(\dfrac{8}{9}\)+\(\dfrac{1}{9}\).\(\dfrac{2}{13}\)+\(\dfrac{1}{9}\).\(\dfrac{11}{13}\)
d) \(\dfrac{3}{4}\).\(\dfrac{8}{9}\).\(\dfrac{15}{16}\). ... .\(\dfrac{9999}{10000}\)
e) \(\dfrac{3}{1.4}\)+\(\dfrac{3}{4.7}\)+\(\dfrac{3}{7.10}\)+...+\(\dfrac{3}{97.100}\)
*Lưu ý: Mong các anh chị trình bày chi tiết để em có thể hiểu bài, em xin các anh chị đừng viết mỗi kết quả xong em chả biết một cái gì ;-;
a: =-21/36-3/36=-24/36=-2/3
b: =43/12*1/2+5/24=43/24+5/24=2
c: =8/9+1/9=1
e: =1-1/4+1/4-1/7+...+1/97-1/100
=1-1/100=99/100
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19 tháng 6 2023 lúc 22:18
Tính tổng S=\(\sqrt{0,49}+\sqrt{\dfrac{1}{9}}-\sqrt{\dfrac{25}{4}}\)
`@` `\text {Ans}`
`\downarrow`
\(S=\sqrt{0,49}+\sqrt{\dfrac{1}{9}}-\sqrt{\dfrac{25}{4}}\)
`S=0,7 + 1/3 - 5/2`
`S=31/30 - 5/2 = -22/15`
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