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(x-20) + (x-19) + (x-18) + ... + 99 + 100 + 101
= 101
<=> (x-20) + (x-19) + (x-18) + ... + 99 + 100
= 0
<=> (x-20) + (x-19) + (x-18) + ... + (x-1) + x + (x+1) + ... + 100
= 0
VT là tổng của 100-(x-20)+1 = 121-x số nguyên liên tiếp
Trung bình cộng của 121-x số nguyên đó là
[(x-20) + 100] / 2
= (80+x)/2
---> (121-x).(80+x)/2 = 0
---> x = 121 và x = -80
\(\left(\frac{1}{1.51}+\frac{1}{2.52}+\frac{1}{3.53}+...+\frac{1}{10.60}\right).x=\frac{1}{1.11}+\frac{1}{2.12}+...+\frac{1}{50.60}\)
\(\Leftrightarrow\left(\frac{50}{1.51}+\frac{50}{2.52}+...+\frac{50}{10.60}\right).x=5.\left(\frac{10}{1.11}+\frac{10}{2.12}+...+\frac{10}{50.60}\right)\)
\(\Leftrightarrow\left(1-\frac{1}{51}+\frac{1}{2}-\frac{1}{52}+...+\frac{1}{10}-\frac{1}{60}\right).x=5.\left(1-\frac{1}{11}+\frac{1}{2}-\frac{1}{12}+...+\frac{1}{50}-\frac{1}{60}\right)\)
\(\Leftrightarrow\left[\left(1+\frac{1}{2}+...+\frac{1}{10}\right)-\left(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{60}\right)\right].x=5.\left[\left(1+\frac{1}{2}+...+\frac{1}{10}\right)-\left(\frac{1}{51}+\frac{1}{52}+..+\frac{1}{60}\right)\right]\)
\(\Leftrightarrow x=5\)
a.
(x-20) + (x-19) + (x-18) + ... + 99 + 100 + 101 = 101
<=> (x-20) + (x-19) + (x-18) + ... + 99 + 100 = 0
<=> (x-20) + (x-19) + (x-18) + ... + (x-1) + x + (x+1) + ... + 100 = 0 VT là tổng của 100-(x-20)+1 = 121-x số nguyên liên tiếp
Trung bình cộng của 121-x số nguyên đó là
[(x-20) + 100] / 2 = (80+x)/2
=> (121-x).(80+x)/2 = 0
=> x = 121 và x = -80
a,\(\dfrac{x+1}{65}+\dfrac{x+2}{64}=\dfrac{x+3}{63}+\dfrac{x+4}{62}\)
\(\Rightarrow\dfrac{x+1}{65}+1+\dfrac{x+2}{64}+1=\dfrac{x+3}{63}+1+\dfrac{x+4}{62}\)
\(\Rightarrow\dfrac{x+1+65}{65}+\dfrac{x+2+64}{64}=\dfrac{x+3+63}{63}+\dfrac{x+4+62}{62}\)
\(\Rightarrow\dfrac{x+66}{65}+\dfrac{x+66}{64}-\dfrac{x+66}{63}-\dfrac{x+66}{62}=0\)
\(\Rightarrow\left(x+66\right)\left(\dfrac{1}{65}+\dfrac{1}{64}-\dfrac{1}{63}-\dfrac{1}{62}\right)=0\)
\(\Rightarrow x+66=0\) ( vì \(\dfrac{1}{65}+\dfrac{1}{64}-\dfrac{1}{63}-\dfrac{1}{62}>0\) )
\(\Rightarrow x=-66\)
\(b,\dfrac{x-12}{77}+\dfrac{x-11}{78}=\dfrac{x-74}{15}+\dfrac{x-73}{16}\)
\(\Rightarrow\dfrac{x-12}{77}-1+\dfrac{x-11}{78}-1=\dfrac{x-74}{15}-1+\dfrac{x-73}{16}-1\)
\(\Rightarrow\dfrac{x-12-77}{77}+\dfrac{x-11-78}{78}=\dfrac{x-74-15}{15}+\dfrac{x-73-16}{16}\)
\(\Rightarrow\dfrac{x-89}{77}+\dfrac{x-89}{78}-\dfrac{x-89}{15}-\dfrac{x-89}{16}=0\)
\(\Rightarrow\left(x-89\right)\left(\dfrac{1}{77}+\dfrac{1}{78}-\dfrac{1}{15}-\dfrac{1}{16}\right)=0\)
\(\Rightarrow x-89=0\)
\(\Rightarrow x=89\)
b.
\(\dfrac{x-12}{77}+\dfrac{x-11}{78}=\dfrac{x-74}{15}+\dfrac{x-73}{16}\\ \Rightarrow\left(\dfrac{x-12}{77}-1\right)+\left(\dfrac{x-11}{78}-1\right)=\left(\dfrac{x-74}{15}-1\right)+\left(\dfrac{x-73}{16}-1\right)\\ \Leftrightarrow\dfrac{x-89}{77}+\dfrac{x-89}{78}=\dfrac{x-89}{15}+\dfrac{x-89}{16}\\ \Leftrightarrow\dfrac{x-89}{77}+\dfrac{x-89}{78}-\dfrac{x-89}{15}-\dfrac{x-89}{16}=0\\ \\ \Leftrightarrow\left(x-89\right)\left(\dfrac{1}{77}+\dfrac{1}{78}-\dfrac{1}{15}-\dfrac{1}{16}\right)=0\\ \Leftrightarrow x-89=0\\ \Leftrightarrow x=89\)