Question

M = 2²⁰¹⁰ - (2^{2009 +} 2²⁰⁰⁸ + ... + 2¹ + 2⁰ )

tính

Answer 1

\(\text{đầu tiên bạn phải biết công thức này:}\)

\(a^{n\: }-b^{n\: }=(a-b)\left(a^{n-1\: }+a^{n-2\: }.b+a^{n-3}.b^2+...+b^{n-1}n\right)\)

\(\text{2^2009 +2^2008 + 2 + 1 = (2^2010 - 1)/(2-1) = 2^2010 - 1 }\)

\(\text{2^2010-2^2009-2^2008...-2-1 = 2^2010 - (2^2009 +2^2008 + 2 + 1) =1}\)

Answer 2

\(M=2^{2010}-\left(2^{2009}+2^{2008}+...+2^1+2^0\right)\)

\(2^{2010}-M=1+2+...+2^{2008}+2^{2009}\)

\(2\left(2^{2010}-M\right)=2+2^2+...+2^{2009}+2^{2010}\)

\(2\left(2^{2010}-M\right)-\left(2^{2010}-M\right)=\left(2+2^2+...+2^{2009}+2^{2010}\right)-\left(1+2+...+2^{2008}+2^{2009}\right)\)

\(2^{2010}-M=2^{2010}-1\)

\(M=2^{2010}-2^{2010}-1=-1\)