a) Do M nằm giữa A và B \(\Rightarrow MA+MB=AB\)

Ta có: \(MA=AB-MB=6-3=3\left(cm\right)\)

Mà: \(MA=MB\) vậy M nằm giữa A và B

\(\Rightarrow\) M là trung điểm của AB 

b) Ta có N là trung điểm của MB 

\(\Rightarrow NB=\dfrac{MB}{2}=\dfrac{3}{2}=1,5\left(cm\right)\)

\(\Rightarrow NB+AN=AB\Rightarrow AN=AB-NB=6-1,5=4,5\left(cm\right)\)

Theo dãy số ta có:

Chứ số thứ 1 là: \(1=1\times1\)

Chữ số thứ 2 là: \(2=1\times2\)

Chữ số thứ 3 là: \(3=1\times3\)

......

Vậy chữ số thứ 100 là: \(1\times100=100\)

333...333 (100 số)x333.333 (100 số)

= 3.111...111(100 số 1)x3.111...111 (100 số 1)

= 9.(111...111)² (100 số)

Đặt A₁₀₀=9.(111...111)² (100 số)

Ta thấy :

A₂=11²=121 ⇒121.9=1089

A₃=111²=12321 ⇒12321.9=110889

A₁₀=11..11² (10 chữ số)=1234567900987654321 ⇒A₁₀.9=11111111108888888889

A₁₅=11..11² (15 chữ số)=12345679012345654320987654321 ⇒A₁₀.9=11..1088..89 (14 số 1, 1 sô 0, 14 số 8, 1 số 9)

...............

⇒ A₁₀₀.9=11..1088..89 (99 số 1, 1 sô 0, 99 số 8, 1 số 9)

\(13-32+51-70+89-108+127-146+165\)

\(=\left(13+127\right)-\left(32+108\right)+\left(51+89\right)-\left(70+146-165\right)\)

\(=140-140+140-51\)

\(=140-51\)

\(=89\)

Mong mn giúp mình nhanh nha mik đang cần rất gấp gáp đó chiều nay mik đi học thêm rồi

\(\dfrac{2}{3}\)\(x\) + (\(\dfrac{4}{5}\)\(x\) - \(\dfrac{11}{15}\)) = \(\dfrac{5}{9}\)

\(\dfrac{2}{3}\)\(x\) + \(\dfrac{4}{5}\)\(x\) - \(\dfrac{11}{15}\) = \(\dfrac{5}{9}\)

(\(\dfrac{2}{3}+\dfrac{4}{5}\))\(x\) = \(\dfrac{5}{9}\) + \(\dfrac{11}{5}\)

\(\dfrac{22}{15}\)\(x\)        = \(\dfrac{58}{45}\)

     \(x\)        = \(\dfrac{58}{45}\): \(\dfrac{22}{15}\)

     \(x\)       = \(\dfrac{29}{33}\)

`@` `\text {Ans}`

`\downarrow`

\(\dfrac{2}{3}+\left[\dfrac{4}{5}x-\dfrac{11}{15}\right]=\dfrac{5}{9}\)

`=>`\(\dfrac{4}{5}x-\dfrac{11}{15}=\dfrac{5}{9}-\dfrac{2}{3}\)

`=>`\(\dfrac{4}{5}x-\dfrac{11}{15}=-\dfrac{1}{9}\)

`=>`\(\dfrac{4}{5}x=-\dfrac{1}{9}+\dfrac{11}{15}\)

`=>`\(\dfrac{4}{5}x=\dfrac{28}{45}\)

`=>`\(x=\dfrac{28}{45}\div\dfrac{ 4}{5}\)

`=>`\(x=\dfrac{7}{9}\)

Vậy, `x = 7/9.`

\(\left(\dfrac{3}{4}-3x\right)\cdot\left(1+4x\right)=0\)

`=>`\(\left[{}\begin{matrix}\dfrac{3}{4}-3x=0\\1+4x=0\end{matrix}\right.\)

`=>`\(\left[{}\begin{matrix}3x=\dfrac{3}{4}\\4x=-1\end{matrix}\right.\)

`=>`\(\left[{}\begin{matrix}x=\dfrac{1}{4}\\x=-\dfrac{1}{4}\end{matrix}\right.\)

Vậy, `x \in {1/4; -1/4}.`

\(\dfrac{-2}{7}+\dfrac{2}{3}:x=\dfrac{-1}{5}\)

⇒\(\dfrac{2}{3}:x=\dfrac{-1}{5}-\dfrac{-2}{7}=\dfrac{3}{35}\)

⇒\(x=\dfrac{2}{3}:\dfrac{3}{35}=\dfrac{70}{9}\)

`@` `\text {Ans}`

`\downarrow`

\(\dfrac{1}{3}\div x=-\dfrac{5}{7}-\left(-\dfrac{2}{3}\right)\)

`=>`\(\dfrac{1}{3}\div x=-\dfrac{5}{7}+\dfrac{2}{3}\)

`=>`\(\dfrac{1}{3}\div x=-\dfrac{1}{21}\)

`=>`\(x=\dfrac{1}{3}\div\left(-\dfrac{1}{21}\right)\)

`=> x = -7`

Vậy `x = -7.`

1/3:x=-5/7- -2/3
1/3:x=-1/21
      x=1/3: (-1/21)
      x=-7

,làm ơn giúp mik với ah

\(\left(1+\dfrac{2}{3}\right).\left(1+\dfrac{2}{4}\right).\left(1+\dfrac{2}{5}\right)....\left(1+\dfrac{2}{2020}\right).\left(1+\dfrac{2}{2021}\right)\)

= \(\dfrac{5}{3}.\dfrac{6}{4}.\dfrac{7}{5}.\dfrac{8}{6}.\dfrac{9}{7}....\dfrac{2022}{2020}.\dfrac{2023}{2021}\)

= \(\dfrac{1}{3}.\dfrac{1}{4}.2022.2023\)

= \(\dfrac{337.2023}{2}\)

= \(\dfrac{\text{681751}}{2}\)

`@` `\text {Ans}`

`\downarrow`

\(\dfrac{7}{4}x+\dfrac{2}{3}=\dfrac{5}{6}\)

`\Rightarrow`\(\dfrac{7}{4}x=\dfrac{5}{6}-\dfrac{2}{3}\)

`\Rightarrow`\(\dfrac{7}{4}x=\dfrac{1}{6}\)

`\Rightarrow`\(x=\dfrac{1}{6}\div\dfrac{7}{4}\)

`\Rightarrow` `x=2/21`

Vậy, `x = 2/21.`

\(\dfrac{7}{4}.x+\dfrac{2}{3}=\dfrac{5}{6}\)

⇒\(\dfrac{7}{4}.x=\dfrac{5}{6}-\dfrac{2}{3}=\dfrac{1}{6}\)

⇒\(x=\dfrac{1}{6}:\dfrac{7}{4}=\dfrac{1}{6}.\dfrac{4}{7}=\dfrac{2}{21}\)