Đáp án:
Vậy $n=1025$
Giải thích các bước giải:
Đặt $A = 2.2^2+3.2^3+4.2^4+...+ (n-1).2^{n-1}+n.2^n$
$=>2A= 2.(2.2^2+3.2^3+4.2^4+...+ (n-1).2^{n-1}+n.2^n )$
$=>2A=2.2^3+3.2^4+4.2^5+...+ (n-1).2^{n}+n.2^{n+1}$
$=>2A-A=(2.2^3+3.2^4+4.2^5+...+ (n-1).2^{n}+n.2^{n+1})-(2.2^2+3.2^3+4.2^4+...+ (n-1).2^{n-1}+n.2^n)$
$=>2A-1.A=(2.2^3-3.2^3)+(3.2^4-4.2^4)+...+ [(n-1).2^{n}-n.2^n]+(n.2^{n+1}-2.2^2)$
$=>(2-1).A=-2^3+(-2^4)+...+ (-2^n)+(n.2^{n+1}-2.2^2)$
$=>A=-2^3+(-2^4)+...+ (-2^n)+(n.2^{n+1}-2.2^2)$
Đặt $B=-2^3+(-2^4)+...+ (-2^n)$
$2B=-2^4+(-2^5)+...+ (-2^{n+1})$
$2B-B=[-2^4+(-2^5)+...+ (-2^{n+1})]-[-2^3+(-2^4)+...+ (-2^n)]$
$B= (-2^{n+1})-(-2^3)$
$B= -2^{n+1}+2^3$
$=>A=(-2^{n+1}+2^3)+(n.2^{n+1}-2.2^2)$
$=>A=-2^{n+1}+n.2^{n+1}+(2^3-2.2^2)$
$=>A=-1.(2^{n+1})+n.2^{n+1}$
$=>A=(-1+n).(2^{n+1})$
$=>(-1+n).(2^{n+1})=2^{n+11}$
$=>-1+n=2^{n+11}:(2^{n+1})$
$=>-1+n=2^{10}$
$=>n=2^{10}+1$
$=>n=1025$
Vậy $n=1025$
