a: \(A=9+99+999+\cdots+999\ldots9\) (10 chữ số 9)
\(=10-1+100-1+1000-1+\cdots+10^{10}-1\)
\(=\left(10+10^2+10^3+\cdots+10^{10}\right)-10\)
Đặt \(E=10+10^2+\cdots+10^{10}\)
=>\(10E=10^2+10^3+\cdots+10^{11}\)
=>\(10E-E=10^2+10^3+\cdots+10^{11}-10-10^2-\cdots-10^{10}\)
=>\(9E=10^{11}-10\)
=>\(E=\frac{10^{11}-10}{9}\)
=>\(A=E-10=\frac{10^{11}-10}{9}-10=\frac{10^{11}-100}{9}\)
b: \(B=1+11+111+\cdots+111\ldots1\) (10 chữ số 1)
\(=\frac19\left(9+99+999+\cdots+999\ldots9\right)\) (10 chữ số 9)
\(=\frac19\cdot A=\frac19\cdot\frac{10^{11}-100}{9}=\frac{10^{11}-100}{81}\)
c: \(C=5+55+555+\cdots+555\ldots5\) (10 chữ số 5)
\(=\frac59\left(9+99+999+\cdots+999\ldots9\right)\) (10 chữ số 9)
\(=\frac59\cdot A=\frac59\cdot\frac{10^{11}-100}{9}=\frac{5\cdot10^{11}-500}{81}\)
d: \(C=4+44+444+\cdots+444\ldots4\) (10 chữ số 4)
\(=\frac49\left(9+99+999+\cdots+999\ldots9\right)\) (10 chữ số 9)
\(=\frac49\cdot A=\frac49\cdot\frac{10^{11}-100}{9}=\frac{4\cdot10^{11}-400}{81}\)
