Question

[Nguyễn Quang Minh]

A= 2+3.2^2+4.2^3+5.2^4+....+61.2^60
Rút gọn A

Answer 1

[Nguyễn Việt Lâm]

\(A=3.2^2+4.2^3+...+60.2^{59}+61.2^{60}\)

\(\Rightarrow2A=3.2^3+4.2^4+...+60.2^{60}+61.2^{61}\)

\(\Rightarrow A-2A=3.2^2+2^3+2^4+...++2^{60}-61.2^{61}\)

\(\Rightarrow-A=5+1+2^1+2^2+2^3+...+2^{60}-61.2^{61}\)

\(\Rightarrow-2A=10+2^1+2^2+2^3+...+2^{61}-61.2^{62}\)

\(\Rightarrow-A-\left(-2A\right)=-4-62.2^{61}+61.2^{62}\)

\(\Rightarrow A=-4+2^{61}\left(-62+61.2\right)\)

\(\Rightarrow A=60.2^{61}-4\)

Answer 2

[Akai Haruma]

Lời giải:

$A=2+3.2^2+4.2^3+5.2^4+....+61.2^{60}$

$2A=4+3.2^3+4.2^4+5.2^5+....+61.2^{61}$

$\Rightarrow A=2A-A=2-12-(2^3+2^4+2^5+....+2^{60})+61.2^{61}$

$\Rightarrow A=61.2^{61}-(2^3+2^4+2^5+...+2^{60})-10$

$2A=61.2^{62}-(2^4+2^5+2^6+...+2^{61})-20$

$2A-A=[61.2^{62}-(2^4+2^5+2^6+...+2^{61})-20]-[61.2^{61}-(2^3+2^4+2^5+...+2^{60})-10]$

$\Rightarrow A=61.2^{62}-61.2^{61}-2^{61}-20+2^3+10$

$\Rightarrow A=61.2^{62}-62.2^{61}-2$

$\Rightarrow A=2^{61}(61.2-62)-2=60.2^{61}-2$