Gọi 5+55+555+55..55(20 chữ số 5 ) = A
Khi đó:
9/5.A = 9/5.(5+55+555+55...55(20 chữ số A))
\(\dfrac{9}{5}.A=\dfrac{9}{5}.5+\dfrac{9}{5}.55+\dfrac{9}{5}.555+...+\dfrac{9}{5}.55...55\left(20chữsố5\right)\)
\(\dfrac{9}{5}.A=9+99+999+...+99...99\left(20chữsố9\right)\)
\(\dfrac{9}{5}.A=\left(10-1\right)+\left(100-1\right)+\left(1000-1\right)+...+\left(100...00\left(20chữsố0\right)-1\right)\)
\(\dfrac{9}{5}.A=\left(10+100+1000+...+100...00\left(20chữsố0\right)\right)-\left(1+1+1+...+1\left(20số1\right)\right)\)
\(\dfrac{9}{5}.A=11...110\left(20chữsố1\right)-20\)
\(\dfrac{9}{5}.A=11...11090\left(18chữsố1\right)\)
\(A=11...11090\left(18chữsố1\right):\dfrac{9}{5}\)
\(A=55...55450\left(18chữsố5\right):9\)
\(A=\dfrac{55...5500...00\left(9chữsố5,12chữsố0\right)}{9}+\dfrac{55...55000\left(9chữsố5\right)}{9}+\dfrac{450}{9}\)
\(A=6172839500...00\left(12chữsố0\right)+61728395000+50\)
\(A=6172839561728395050\)
Vậy 5+55+555+...+55...55(20 chữ số 5) = 6172839561728395050
Kết quả hơi lẻ, mình không rõ có đúng không.
