Question

B1:

Cho \(A=1+2+2^2+2^3+...+2^{10}\)

So sánh \(A\) với \(2^{11}\)

B2:

Cho \(B=2.2^2+3.2^3+4.2^4+...+10.2^{10}\)

So sánh \(B\) với \(2^{14}\)

Answer 1

_{\(A=1+2+2^2+2^3+...+2^{10}\)}

_{\(2A=2+2^2+2^3+2^4+...+2^{11}\)}

^{\(2A-A=\left(2+2^2+2^3+2^4+...+2^{11}\right)-\left(1+2+2^2+2^3+...+2^{10}\right)\)}

^{\(A=2^{11}-1< 2^{11}\)}

^{\(B=2.2^2+3.2^3+4.2^4+...+10.2^{10}\)\(2B=2.2^3+3.2^4+4.2^5+...+10.2^{11}\)\(2B-B=\left(2.2^3-3.2^3\right)+\left(3.2^4-4.2^4\right)+...+\left(9.2^{10}-10.2^{10}\right)+10.2^{11}-2.2^2\)\(B=2^3\left(2-3\right)+2^4\left(3-4\right)+...+2^{10}\left(9-10\right)+10.2^{11}-2.2^2\)\(B=-2^3-2^4-....-2^{10}+10.2^{11}-2^3\)}

_{\(B=-\left(2^3+2^4+...+2^{10}\right)+10.2^{11}-2^3\)}

^{\(B=-\left(2^{11}-2^3\right)+10.2^{11}-2^3\)}

_{\(B=-2^{11}+2^3+10.2^{11}-2^3\)}

_{\(B=9.2^{11}\)}

Ta cần so sánh: _\(9.2^{11}\) và _\(2^{14}\)

Hay _\(9\) và _\(2^3\)

_{\(9>8=2^3\Leftrightarrow B>2^{14}\)}