A= 2+3.2^2+4.2^3+5.2^4+....+61.2^60
Rút gọn A
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HK
3
28 tháng 2 2019
\(2A=2.2^3+3.2^4+4.2^5+...+100.2^{101}\)
=> \(2A-A=100.2^{101}-\left(2^{100}+2^{99}+...+2^4+2^3\right)-2.2^2\)
Đặt \(B=2^3+2^4+...+2^{100}\Rightarrow2B=2^4+2^5+...+2^{101}\)
=> \(2B-B=2^{101}-2^3\Rightarrow B=2^{101}-2^3\)
=> \(2A-A=100.2^{101}-\left(2^{101}-2^3\right)-2.2^2\)
=> \(A=\left(100.2^{101}-2^{101}\right)+2^3-2^3\)=\(99.2^{101}\)
NT
0
AN
1
17 tháng 11 2021
B=2.22+3.23+4.24+......+10.210
Hãy so sánh B với 214
Nhanh nhất, cụ thể và đúng nhất, 10k
DM
1
BC
1
KN
1 tháng 8 2019
Đặt \(A=2.2^2+3.2^3+4.2^4+5.2^5+...+n.2^n\)
\(\Rightarrow2A=2.2^3+3.2^4+4.2^5+5.2^6+...+n.2^{n+1}\)
\(\Rightarrow2A-A=2.2^3+3.2^4+4.2^5+5.2^6+...+n.2^{n+1}\)
\(-2.2^2-3.2^3-4.2^4-5.2^5-...-n.2^n\)
\(A=n.2^{n+1}-2^3-\left(2^3+2^4+...+2^n\right)\)
Đặt \(M=\left(2^3+2^4+...+2^n\right)\)
\(\Rightarrow2M=\left(2^4+2^5+...+2^{n+1}\right)\)
\(\Rightarrow M=2^{n+1}-2^3\)
\(\Rightarrow A=n.2^{n+1}-2^3-2^{n+1}+2^3\)
\(\Rightarrow A=\left(n-1\right)2^{n+1}=2^{n+10}\)
\(\Rightarrow\left(n-1\right)=2^9\)
\(\Rightarrow n=513\)
BC
1
1 tháng 8 2019
Đặt \(A=2.2^2+3.2^3+4.2^4+...+n.2^n=2^{n+10}\)
\(\Rightarrow2A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}\)
\(\Rightarrow2A-A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}-2.2^2-3.2^3-4.2^4-...-n.2^n\)
\(\Leftrightarrow A=-2.2^2+\left(2.2^3-3.2^3\right)+\left(3.2^4-4.2^4\right)+...+[\left(n-1\right)2^n-n.2^n]+n.2^{n+1}\)
\(\Leftrightarrow A=-2.2^2-2^3-2^4-...-2^n+n.2^{n+1}\)
\(\Leftrightarrow A=-2^3-\left(2^4-2^3\right)-\left(2^5-2^4\right)-...-\left(2^{n+1}-2^n\right)+n.2^{n+1}\)
\(\Leftrightarrow A=-2^3-2^4+2^3-2^5+2^4-...-2^{n+1}+2^n+n.2^{n+1}\)
\(\Leftrightarrow A=-2^{n+1}+n.2^{n+1}\)
\(\Leftrightarrow A=2^{n+1}\left(n-1\right)\)
Mà \(A=2^{n+10}=2^{n+1}.2^9=2^{n+1}.512\)
\(\Rightarrow n-1=512\)
\(\Rightarrow n=513\)
LS
1
\(A=3.2^2+4.2^3+...+60.2^{59}+61.2^{60}\)
\(\Rightarrow2A=3.2^3+4.2^4+...+60.2^{60}+61.2^{61}\)
\(\Rightarrow A-2A=3.2^2+2^3+2^4+...++2^{60}-61.2^{61}\)
\(\Rightarrow-A=5+1+2^1+2^2+2^3+...+2^{60}-61.2^{61}\)
\(\Rightarrow-2A=10+2^1+2^2+2^3+...+2^{61}-61.2^{62}\)
\(\Rightarrow-A-\left(-2A\right)=-4-62.2^{61}+61.2^{62}\)
\(\Rightarrow A=-4+2^{61}\left(-62+61.2\right)\)
\(\Rightarrow A=60.2^{61}-4\)
Lời giải:
$A=2+3.2^2+4.2^3+5.2^4+....+61.2^{60}$
$2A=4+3.2^3+4.2^4+5.2^5+....+61.2^{61}$
$\Rightarrow A=2A-A=2-12-(2^3+2^4+2^5+....+2^{60})+61.2^{61}$
$\Rightarrow A=61.2^{61}-(2^3+2^4+2^5+...+2^{60})-10$
$2A=61.2^{62}-(2^4+2^5+2^6+...+2^{61})-20$
$2A-A=[61.2^{62}-(2^4+2^5+2^6+...+2^{61})-20]-[61.2^{61}-(2^3+2^4+2^5+...+2^{60})-10]$
$\Rightarrow A=61.2^{62}-61.2^{61}-2^{61}-20+2^3+10$
$\Rightarrow A=61.2^{62}-62.2^{61}-2$
$\Rightarrow A=2^{61}(61.2-62)-2=60.2^{61}-2$