`S = 1 + 5 + 5^{2} +...+ 5^{10}`
`5S = 5 . (5^{0} + 5 + 5^{2} +...+ 5^{10})`
`5S = 5^{1} + 5^{2} + 5^{3} +...+ 5^{11}`
`5S - S= (5^{1} + 5^{2} + 5^{3} +...+ 5^{11}) - (1 + 5 + 5^{2} +...+ 5^{10})`
`4S= 5^{11} - 1`
`S= {5^{11} - 1}/4`
_______________________
`45 + 1 = 5^{n}`
`46 = 5^{n}`
$\Rightarrow$ $\text{Vô lí}$
$\text{Vậy}$ `n \in \emptyset`