Question

[Hồ Phan Phú Thịnh]

Tính  a/           A =2 +22 +222 +.........+22........2

                                                              50 số 2

b/  1/1.2.3 +1/2.3.3 + ......+1/98.99.100

Answer 1

[Kiệt Nguyễn]

đề hơi sai, mk sửa lại nhé

Đặt \(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{98.99.100}\)

\(\Leftrightarrow2A=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{98.99.100}\)

\(\Leftrightarrow2A=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{98.99}-\frac{1}{99.100}\)

\(\Leftrightarrow2A=\frac{1}{1.2}-\frac{1}{99.100}\)

\(\Leftrightarrow2A=\frac{4950}{9900}-\frac{1}{9900}\)

\(\Leftrightarrow2A=\frac{4949}{9900}\)

\(\Leftrightarrow A=\frac{4949}{9900}\div2\)

\(\Leftrightarrow A=\frac{9898}{9900}=\frac{4949}{4950}\)

Answer 2

[Kiệt Nguyễn]

\(A=2+22+...+222222222...2222\left(\text{50 chữ số 2}\right)\)

​\(\Leftrightarrow A=2\left(1+11+...+111111111...1111\right)\)​

\(\Leftrightarrow A=\frac{2}{9}\left(9+99+...+9999999999...99\right)\)

\(\Leftrightarrow A=\frac{2}{9}\left[\left(10-1\right)+\left(10^2-1\right)+...+\left(10^{50}-1\right)\right]\)

\(\Leftrightarrow A=\frac{2}{9}\left[\left(10+10^2+...+10^{50}\right)-50\right]\)

Đặt \(B=\left[\left(10+10^2+...+10^{50}\right)\right]\)

\(\Leftrightarrow10B=\left[\left(10^2+10^3+...+10^{51}\right)\right]\)

\(\Leftrightarrow10B-B=\left[\left(10^2+10^3+...+10^{51}\right)\right]-10-10^2-...-10^{50}\)

\(\Leftrightarrow10B-B=10^{51}-10\)

\(\Rightarrow A=\frac{2}{9}\left(10^{51}-10-50\right)\)

\(\Rightarrow A=\frac{2}{9}\left(10^{51}-60\right)\)