A=2+2^2+2^3+...+2^30 chứng tỏ rằng a+2=31

Bạn tham khảo   :

$A = 2+2^2+2^3+...+2^{30}$

$2A = 2^2 + 2^3 + 2^4 + ... 2^{31}$

$2A - A = ( 2^2 + 2^3 + 2^4 + ... 2^{31}) - ( 2+2^2+2^3+...+2^{30})$

$A        = (2^2 - 2^2) + (2^3 - 2^3) + (2^4 - 2^4) + ... + (2^{31} -2 )$ 

$A       = 2^{31} - 2$

Ta có :

$2^{31} - 2 = 2^{31} + 2 - 2$

Mà $A + 2 = 2^{31} -2+2  = 2^{31}$

Vậy $A+2 = 2^{31}$ 

Ta có

$A = 2 + 2^2 + \cdots + 2^{30}$

$2A = 2^2 + 2^3 + \cdots + 2^{30} + 2^{31}$

Vậy

$A = 2A - A = (2^2 + 2^3 + \cdots + 2^{30} + 2^{31}) - (2 + 2^2 + \cdots + 2^{30}) = 2^{31} - 2$

Khi đó, ta có

$A+2 = 2^{31}-2 + 2 = 2^{31}$