`1` + `5` + $5^{2}$ + ..... + $5^{2000}$
Đặt A = `1` + `5` + $5^{2}$ + ..... + $5^{2000}$
5A = `5` + $5^{2}$ + $5^{3}$ + ..... + $5^{2000}$ + $5^{2001}$
5A - A =( `5` + $5^{2}$ + $5^{3}$ + ..... + $5^{2000}$ + $5^{2001}$ ) - ( `1` + `5` + $5^{2}$ + ..... + $5^{2000}$ )
4A = $5^{2001}$ - `1`
A = `($5^{2001}$ - `1`)``/`4`
Vậy `1` + `5` + $5^{2}$ + ..... + $5^{2000}$ =($5^{2001}$ - `1`)``/`4`