\(S=1-4+4^2-4^3+...4^{100}\)

\(\Rightarrow4S=4-4^2+4^3-4^4+....+4^{101}\)

\(\Rightarrow4S-S=\left(4-4^2+4^3-4^4+...+4^{101}\right)-\left(1-4+4^2-4^3+...+4^{100}\right)\)\(\Rightarrow3S=4^{101}-1\)

\(\Rightarrow S=\frac{4^{101}-1}{3}\)

1.B

2.B

3.C

4.B

1. He made me (do) do it all over again

2. She can (sing) sing quite well

3.He will be able (swim) to swim very soon

4. I used to ( live) live in a caravan

5. You ought (go) to go today. It may (rain) rain tomorow

I. Rewrite the following sentences, keep their meaning.

1. What was your height last year? => How heavy were you last year ?

2. would you like to go to the cafeteria for lunch ?

=> What about going to the cafeteria for lunch ?

\(A=\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+...+\frac{1}{4950}\)

\(=\frac{2}{20}+\frac{2}{30}+\frac{2}{42}+...+\frac{2}{9900}\)

\(=\frac{2}{4.5}+\frac{2}{5.6}+\frac{2}{6.7}+...+\frac{2}{99.100}\)

\(=2\left(\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+...+\frac{1}{99.100}\right)\)

\(=2\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{99}-\frac{1}{100}\right)\)\(=2\left(\frac{1}{4}-\frac{1}{100}\right)\)

\(=2.\frac{6}{25}\)

\(=\frac{12}{25}\)

2c)

Ta có: \(25-y^2\le25\Rightarrow8\left(x-2012\right)^2\le25\)

\(\Rightarrow\left(x-2012\right)^2\le3\)

\(\Rightarrow\left[\begin{matrix}\left(x-2012\right)^2=0\\\left(x-2012\right)^2=1\end{matrix}\right.\)

\(\Rightarrow\left[\begin{matrix}x-2012=0\\\left[\begin{matrix}x-2012=1\\x-2012=-1\end{matrix}\right.\end{matrix}\right.\)

\(\Rightarrow\left[\begin{matrix}x=2012\\\left[\begin{matrix}x=2013\\x=2011\end{matrix}\right.\end{matrix}\right.\)\(\Rightarrow\left[\begin{matrix}y=5\\\left[\begin{matrix}y=\sqrt{17}\\y=\sqrt{17}\end{matrix}\right.\end{matrix}\right.\)(loại)

Vậy x=2012,y=5

Từ \(a\le b+1\le c+2\Rightarrow a+b+1+c+2\le3\left(c+2\right)\)\(\Rightarrow a+b+c+3\le3c+6\)

Mà a+b+c=1

\(\Rightarrow4\le3c+6\)

\(\Rightarrow-2\le3c\)

\(\Rightarrow c\ge-\frac{2}{3}\)

Dấu ''='' xảy ra khi \(c=\frac{-2}{3}\)

Vậy c nhỏ nhất khi \(c=\frac{-2}{3}\)

Ở quê các bạn chắc khác mình rùi kiến đen hiền ko cắn người kiến bác hồ còn kiến lửa là kiến trung quốc vì nó cắn người hihi

Ta có:\(49-y^2\le49\Rightarrow12\left(x-2001\right)^2\le49\)

\(\Rightarrow\left(x-2001\right)^2\le4\)

\(\Rightarrow\left(x-2001\right)^2\in\left\{0;1;4\right\}\)

\(\Rightarrow x-2001\in\left\{0;1;2\right\}\)

\(\Rightarrow x\in\left\{2001;2002;2003\right\}\)