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S=2.2^2+3.2^3+...+n.2^n=2^{n+11}
S=2S-S=(2.2^3+3.2^4+4.2^5+...+n.2^{n+1})-(2.2^2+3.2^3+4.2^4+...+n.2^n)
S=n.2^{n+1}-2^3-(2^3+2^4+...+2^{n-1}+2^n)
Dat T=2^3+2^4+...+2^{n-1}+2^n
Ta tinh dc: T=2T-T=2^{n-1}-2^3
S=n.2^{n+1}-2^3-2^{n-1}+2^3=(n-1).2^{n+1}
=> (n-1).2^{n+1}=n^{n+11}
=> n-1=2^{10}
=> n=2^{10}+1
=> n=1024+1
=> n = 1025
Đặt \(A=2.2^2+3.2^3+4.2^4+...+n.2^n\)
\(\Leftrightarrow2A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}\)
\(\Leftrightarrow2A-A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}-\left(2.2^2+3.2^3+4.2^4+...+n.2^n\right)\)
\(\Leftrightarrow A=-2.2^2-2^3-2^4-....-2^n+n.2^{n+1}\)
\(\Leftrightarrow A=-2^{n+1}+n.2^{n+1}=\left(n-1\right).2^{n+1}\)
mà \(A=2^{n+11}\) \(\Leftrightarrow\left(n-1\right).2^{n+1}=2^{n+11}\)
\(\Leftrightarrow\left(n-1\right).2^n.2=2^n.2^{11}\)
\(\Leftrightarrow\left(n-1\right)=2^{10}\)
\(\Leftrightarrow n=2^{10}+1\)
2.2n+3.2n+4.2n+5.2n+...+n.2n=2n+10
<=>(2+3+4+5+...+n).2n=210.2n
<=>2+3+4+5+...+n=210
<=>n không tồn tại
Giải:
Đặt \(S=2.2^2+3.2^3+...+n.2^n=2^{n+11}\)
\(S=2S-S=\left(2.2^3+3.2^4+4.2^5+...+n.2^{n+1}\right)-\left(2.2^2+3.2^3+4.2^4+...+n.2^n\right)\)
\(S=n.2^{n+1}-2^3-\left(2^3+2^4+...+2^{n-1}+2^n\right)\)
Đặt \(T=2^3+2^4+...+2^{n-1}+2^n\)
Ta tính được: \(T=2T-T=2^{n-1}-2^3\)
\(\Rightarrow S=n.2^{n+1}-2^3-2^{n-1}+2^3=\left(n-1\right).2^{n+1}\)
\(\Rightarrow\left(n-1\right).2^{n+1}=n^{n+11}\)
\(\Rightarrow n-1=2^{10}\)
\(\Rightarrow n=2^{10}+1\)
\(n=1024+1\)
\(\Rightarrow n=1025\)