`A = 4 + 2^{2} + 2^{3} + ... + 2^{2005}`
`A = 2^{2} + 2^{2} + 2^{3} + ... + 2^{2005}`
`2A = 2 . (2^{2} + 2^{2} + 2^{3} + ... + 2^{2005})`
`2A = 2^{3} + 2^{3} + 2^{4} + ... + 2^{2006}`
`2A - A = (2^{3} + 2^{3} + 2^{4} + ... + 2^{2006}) - (2^{2} + 2^{2} + 2^{3} + ... + 2^{2005})`
`A = 2^{3} + 2^{2006} - 2^{2}`
`A = 2^{3} - 2^{2} + 2^{2006}`
`A = 8 - 4 + 2^{2006}`
`A = 4 + 2^{2006}`
`A = 2^{2} + 2^{2006}`
$\text{Vậy bài toán được chứng minh}$
