Bài 1:
\(S=2^2+4^2+6^2+...+20^2\)
\(=\left(1\cdot2\right)^2+\left(2\cdot2\right)^2+\left(2\cdot3\right)^2+...+\left(2\cdot10\right)^2\)
\(=1\cdot2^2+2^2\cdot2^2+2^2\cdot3^2+...+2^2\cdot10^2\)
\(=2^2\left(1+2^2+3^2+...+10^2\right)\)
\(=4\cdot385=1540\)
Bài 2:
\(A=2^0+2^1+2^2+...+2^{100}\)
\(A=1+2+2^2+...+2^{100}\)
\(2A=2\left(1+2+2^2+...+2^{100}\right)\)
\(2A=2+2^2+2^3+...+2^{101}\)
\(2A=\left(2+2^2+...+2^{101}\right)-\left(1+2+...+2^{100}\right)\)
\(A=2^{101}-1\)
Giải:
\(1.\) \(S=2^2+4^2+6^2+....+20^2\)
\(2^2=\left(1.2\right)^2\)
\(4^2=\left(2.2\right)^2\)
\(...\)
Vế dưới \(= \left(1.2\right)^2 + \left(2.2\right)^2 + ...+ \left(9.2\right)^2+ \left(10.2\right)^2\)
\(= 2^2.(1^2 + 2^2 + 3^2 + ...+ 9^2 + 10^2) \)
\(= 4. 385\)
\(= 1540\)
\(2.\)
\( 2A = 2^1 + 2^2 + 2^3 + 2^4 +...+\)\(2^{2011}\)
\(2A - A = ( 2^1 + 2^2 + 2^3+ 2^4 +...+ 2^{2011} ) - ( 1 + 2^2 + 2^3 +...+ 2^{2010} ) \)
\(\Rightarrow A = 2^{2011} - 1\)
\(S=2^2+4^2+6^2+..........+20^2\)
\(S=\left(1.2\right)^2+\left(2.2\right)^2+\left(2.3\right)^2+........+\left(2.10\right)^2\)
\(S=2^2\left(1^2+2^2+3^2+..........+10^2\right)\)
\(S=2^2.385\)
\(S=1540\)
\(A=2^0+2^1+2^2+.....+2^{100}\)
\(A=1+2+2^2+........+2^{100}\)
\(2A=2\left(1+2+2^2+.........+2^{100}\right)\)
\(2A=2+2^2+2^3+.........+2^{101}\)
\(2A-A=\left(2+2^2+2^3+......+2^{101}\right)-\left(1+2+2^2+........+2^{100}\right)\)\(A=2^{101}-1\)
Ta có : \( 1^2+2^2+3^2+...+10^2=385\)
\(\Rightarrow\) \(2^2( 1^2+2^2+3^2+...+10^2)=385.2^2\)
\(\Leftrightarrow\) \(S=2^2+4^2+6^2...+20^2=385.4\)
\(=> S=1540\)
2. Tính :
\(A=2^0+2^1+2^2+...+2^{100}\\ \Rightarrow2A=2^1+2^2+2^3+...+2^{101}\\ \Rightarrow2A-A=\left(2^0+2^1+2^2+...+2^{100}\right)-\left(2^1+2^2+2^3+...+2^{101}\right)\\ \Rightarrow A=2^{101}-1\)
