Question

Tìm các số x,y sao cho :

a, 2^x + 2^x+1 + 2^x+2 + 2^x+3 + ...+ 2^x+2015 = 2²⁰¹⁹ -8 

b, (2008x + 3y +1 ) (2008^x+2008x + y ) = 225

c, 36 - y² = 8(x-2010)² 

 

Answer 1

a) \(\Rightarrow2^x.1+2^x.2+2^x.3+...+2^x.2015=2^{2019}-2^3\)

\(\Rightarrow2^x.\left(1+2+2^2+2^3+...+2^{2015}\right)=2^3.\left(2^{2016}-1\right)\)

Đặt \(A=1+2+2^2+2^3+...+2^{2015}\)

\(\Rightarrow2.A=2+2^2+2^3+...+2^{2016}\)

\(\Rightarrow A=2^{2016}-1\)

\(\Rightarrow2^x.\left(2^{2016}-1\right)=2^3.\left(2^{2016}-1\right)\)

\(\Rightarrow2^x=2^3\Rightarrow x=3\)

Vậy x=3