\(S=5+5^2+5^3+...........+5^{2004}\)(\(2004\) số hạng)
\(\Leftrightarrow S=\left(5+5^3\right)+\left(5^2+5^4\right)+..........+\left(5^{2001}+5^{2004}\right)\)(\(1007\) số hạng)
\(\Leftrightarrow S=5\left(1+5^3\right)+5^2\left(1+5^3\right)+..........+5^{2001}\left(1+5^3\right)\)
\(\Leftrightarrow S=5.126+5^2.126+..........+5^{2001}.126\)
\(\Leftrightarrow S=126\left(5+5^2+...........+5^{2001}\right)⋮126\)
\(\Leftrightarrow S⋮126\rightarrowđpcm\)
Giải.
Ta có : \(S=5+5^2+5^2+...+5^{2004}\)
\(\Leftrightarrow S=\left(5+5^2+5^3\right)+\left(5^4+5^5+5^6\right)+...+\left(5^{2002}+5^{2003}+5^{2004}\right)\)\(\Leftrightarrow S=\left(5+5^2+5^3\right)+5^3\left(5+5^2+5^3\right)+...+5^{2001}\left(5+5^2+5^3\right)\)\(\Leftrightarrow S=\left(5+5^2+5^3\right)\left(1+5^3+...+5^{2001}\right)\)
\(\Leftrightarrow S=126\left(1+5^3+...+5^{2001}\right)⋮126\)
Vậy \(S⋮126\)(đpcm)