Question

[192837465]

3+33+333+3333+...+33..333

Số 33...333 có 20 chữ số 3)

Giải chi tiết dùm nhé

Answer 1

[Nguyễn Ngọc Anh Minh]

\(A=3\left(1+11+111+...+11...111\right)=\)

\(=\frac{3}{9}\left(9+99+999+...+99...999\right)=\)

\(=\frac{1}{3}\left[\left(10-1\right)+\left(10^2-1\right)+\left(10^3-1\right)+...+\left(10^{20}-1\right)\right]=\)

\(=\frac{1}{3}\left[\left(10+10^2+10^3+...+10^{20}\right)-20\right]\)

Đặt \(B=10+10^2+10^3+...10^{20}\)

\(\Rightarrow10B=10^2+10^3+10^4+...+10^{21}\)

\(9B=10B-B=10^{21}-10\Rightarrow B=\frac{10^{21}-10}{9}\)

\(\Rightarrow A=\frac{1}{3}\left(\frac{10^{21}-10}{9}-20\right)\)