$\text{Đáp án + Giải thích các bước giải:}$
`A=7^{1}+7^{2}+7^{3}+7^{4}+...+7^{2010}`
`=>A=(7^{1}+7^{2})+(7^{3}+7^{4})+...+(7^{2009}+7^{2010})`
`=>A=7(1+7)+7^{3}(1+7)+...+7^{2009}(1+7)`
`=>A=7.8+7^{3}.8+...+7^{2009}.8`
`=>A=8.(7+7^{3}+...+7^{2009})\vdots 8`
`-------------`
`A=7^{1}+7^{2}+7^{3}+...+7^{2010}`
`=>A=(7^{1}+7^{2}+7^{3})+...+(7^{2008}+7^{2009}+7^{2010})`
`=>A=7(1+7+7^{2})+....+7^{2008}(1+7+7^{2})`
`=>A=7.57+....+7^{2008}.57`
`=>A=57.(7+...+7^{2008})\vdots 57`