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Từ \(\frac{x-2016}{100}\rightarrow\frac{x-2}{2114}\) có tất cả \(1008\) số.
Ta có: \(\frac{x-2016}{100}+\frac{x-2014}{102}+...+\frac{x-2}{2114}=1008\)
\(\Leftrightarrow\frac{x-2016}{100}+\frac{x-2014}{102}+...+\frac{x-2}{2114}-1008=1008-1008\)
\(\Leftrightarrow\left(\frac{x-2016}{100}-1\right)+\left(\frac{x-2014}{102}-1\right)+...+\left(\frac{x-2}{2114}-1\right)=0\) (*)
\(\Leftrightarrow\frac{x-2116}{100}+\frac{x-2116}{102}+...+\frac{x-2116}{2114}=0\)
\(\Leftrightarrow\left(x-2116\right)\left(\frac{1}{100}+\frac{1}{102}+...+\frac{1}{2114}\right)=0\)
Vì \(\frac{1}{100}+\frac{1}{102}+...+\frac{1}{2114}\ne0\)
Nên từ pt (*) \(\Leftrightarrow x-2116=0\Leftrightarrow x=2116\)
Vậy...
\(\frac{x+2}{327}+\frac{x+3}{326}+\frac{x+4}{325}+\frac{x+5}{324}+\frac{x+349}{5}=0.\)
\(1+\frac{x+2}{327}+1+\frac{x+3}{326}+1+\frac{x+4}{325}+1+\frac{x+5}{324}-4+\frac{x+349}{5}=0\)
\(\frac{x+329}{327}+\frac{x+329}{326}+\frac{x+329}{325}+\frac{x+329}{324}+\frac{x+329}{5}=0\)
\(\left(x+329\right).\left(\frac{1}{327}+\frac{1}{326}+\frac{1}{325}+\frac{1}{324}+\frac{1}{5}\right)=0\)
\(\Rightarrow x+329=0\)
\(\Rightarrow x=-329\)
Study well
\(\Rightarrow\frac{x+2}{327}+1+\frac{x+3}{326}+1+\frac{x+4}{325}+1+\frac{x+5}{324}+1+\frac{x+349}{5}-4=0\)
\(\Rightarrow\frac{x+329}{327}+\frac{x+329}{326}+\frac{x+329}{325}+\frac{x+329}{324}+\frac{x+329}{5}=0\)
\(\Rightarrow\left(x+329\right).\left(\frac{1}{327}+\frac{1}{326}+\frac{1}{325}+\frac{1}{324}+\frac{1}{5}\right)=0\)
Mà \(\left(\frac{1}{327}+\frac{1}{326}+\frac{1}{325}+\frac{1}{324}+\frac{1}{5}\right)\ne0\)
Nên \(x+329=0\Rightarrow x=-329\)
Vậy \(x=-329\)
Chúc bạn học tốt !!!
Câu 1 : Đặt A = 1.2.3 + 2.3.4 + ... + 111.112.113
=> 4A = 1.2.3.4 + 2.3.4.4 + ... + 111.112.113.4
= 1.2.3.4 + 2.3.4.(5 - 1) + .... + 111.112.113.(114 - 110)
= 1.2.34 + 2.3.4.5 - 1.2.3.4 + ... + 111.112.113.114 - 110.111.112.113
= 111.112.113.114
=> A = 111.113.114.28 = 40 037 256
Câu 2 Đặt A = 1.2 + 2.3 + 3.4 + ... + 277.278
=> 3A = 1.2.3 + 2.3.3 + 3.4.3 + ... + 277.278.3
= 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 277.278.(279 - 276)
= 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 277.278.279 - 276.277.278
= 277.278.279
=> A = 7161558
3) Đặt A = 1.4 + 2.5 + ... + 277.280
= 1.(2 + 2) + 2.(2 + 3) + ... + 277.(278 + 2)
= (1.2 + 2.3 + .... + 277.278) + 2(1 + 2 + .... 277)
Đặt B = 1.2 + 2.3 + .... + 277.278
=> 3B = 1.2.3 + 2.3.3 + 3.4.3 + ... + 277.278.3
= 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 277.278.(279 - 276)
= 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 277.278.279 - 276.277.278
= 277.278.279
=> B = 7161558
Khi đó A = B + 2(1 + 2 + .... 277)
= 7161558 + 2.277(277 + 1) : 2
= 7238564
Câu 4 : \(\left(\frac{2^2}{2.4}+\frac{2^2}{4.6}+...+\frac{2^2}{34.36}\right)x-1\frac{1}{6}=1\frac{2}{3}\)
=> \(2\left(\frac{2}{2.4}+\frac{2}{4.6}+...+\frac{2}{34.36}\right)x-\frac{7}{6}=\frac{5}{3}\)
=> \(2\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{34}-\frac{1}{36}\right)x=\frac{17}{6}\)
=> \(\left(\frac{1}{2}-\frac{1}{36}\right)x=\frac{17}{12}\)
=> x = 3
Câu 5 : Đặt A = 1 + 2 + 22 + ... + 29 (1)
=> 2A = 2 + 22 + 23 + ... + 210 (2)
Lấy (2) trừ (1) theo vế ta có :
2A - A = (2 + 22 + 23 + ... + 210) - ( 1 + 2 + 22 + ... + 29)
A = 210 - 1 = 1024 - 1 = 1023
Câu 6 : Đặt A = 12 + 22 + 32 + .... + 1002
= 1.1 + 2.2 + 3.3 + ... + 100.100
= 1.(2 - 1) + 2(3 - 1) + 3(4 - 1) + ... + 100(101 - 1)
= (1.2 + 2.3 + 3.4 + ... + 100.101) - (1 + 2 + 3 + 4 + ... + 100)
Đặt B = 1.2 + 2.3 + 3.4 + ... + 100.101
=> 3B = 1.2.3 + 2.3.3 + 3.4.3 + ... + 100.101.3
= 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 100.101(102 - 99)
= 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + .... + 100.101.102 - 99.100.101
= 100.101.102
=> B = 343400
Khi đó A = B - (1 + 2 + 3 + 4 + ... + 100)
= 343 400 - [100.(100 + 1) : 2]
= 338 350
Ta có: \(\frac{xy}{x+y}=\frac{yz}{y+z}=\frac{zx}{z+x}\)\(\Rightarrow\frac{xyz}{z\left(x+y\right)}=\frac{xyz}{x\left(y+z\right)}=\frac{xyz}{y\left(z+x\right)}\)\(\Rightarrow z\left(x+y\right)=x\left(y+z\right)=y\left(z+x\right)\)\(\Rightarrow zx+zy=xy+xz=yz+xy\)
Ta có: zx + zy = xy + xz => zy = xy => z = x (1)
Ta có: x - z = x - x = 0
\(\frac{x+2}{327}+\frac{x+3}{326}+....+\frac{x+349}{5}=\frac{x+2}{327}+\frac{x+3}{326}+\frac{x+4}{325}+\frac{x+5}{324}+\frac{x+329}{5}+4=\left(\frac{x+2}{327}+1\right)+\left(\frac{x+3}{326}+1\right)+\left(\frac{x+4}{325}+1\right)+\left(\frac{x+5}{324}+1\right)+\frac{x+329}{5}=\frac{x+329}{327}+\frac{x+329}{326}+\frac{x+329}{325}+\frac{x+329}{324}+\frac{x+329}{5}=\left(x+329\right)\left(\frac{1}{327}+\frac{1}{326}+....+\frac{1}{324}+\frac{1}{5}\right)=0\Rightarrow x+329=0\Leftrightarrow x=-329\)
Bạn tham khảo tại đây nhé: Câu hỏi của trần như.
Chúc bạn học tốt!
Đặt 4+6+8+10+...+2012 là A
Ta có: số số hạng A là:(2012-4)/2+1=1005
tổng A là:(2012+4).1005/2=1013040
=1013040.\(\frac{1}{1000}\) .(\(\frac{1}{2}+\frac{3}{4}+\frac{5}{6}\))
=1013,04.(\(\frac{6}{12}+\frac{9}{12}+\frac{10}{12}\))
=1013,04.\(\frac{25}{12}\)
=2110,5
\(\frac{x-2016}{100}+\frac{x-2014}{102}+\frac{x-2016}{104}+...+\frac{x-2}{2114}=1008\)
\(\Rightarrow\frac{x-2016}{100}-1+\frac{x-2014}{102}-1+...+\frac{x-2}{2114}-1=0\)
\(\Rightarrow\frac{x-2116}{100}+\frac{x-2116}{102}+...+\frac{x-2116}{2114}=0\)
\(\Rightarrow\left(x-2116\right)\left(\frac{1}{100}+\frac{1}{102}+...+\frac{1}{2114}\right)=0\)
mà \(\frac{1}{100}+\frac{1}{102}+...+\frac{1}{2114}\ne0\)
\(\Rightarrow x-2116=0\)
\(\Rightarrow x=2116\)
P/s màu mè ghê ha =))
\(\frac{x-2016}{100}+\frac{x-2014}{102}+...+\frac{x-2}{2114}=1008\)
\(=>\frac{x-2016}{100}+\frac{x-2014}{102}+...+\frac{x-2}{2114}-1008=0\)
\(=>\frac{x-2016}{100}-1+\frac{x-2014}{102}-1+...+\frac{x-2}{2114}-1=0\)
\(=>\frac{x-2116}{100}+\frac{x-2116}{102}+...+\frac{x-2116}{2114}=0\)
\(=>\left(x-2116\right).\left(\frac{1}{100}+\frac{1}{102}+...+\frac{1}{2114}\right)=0\)
Do \(\frac{1}{100}+\frac{1}{102}+...+\frac{1}{2114}\ne0\)
\(=>x-2116=0\)
\(=>x=2116\)