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a, A = 1 + 3 + 32 + 33 + ... + 32000
3.A = 3 + 32 + 33+ 33+... + 32001
3A - A = 3 + 32 + 33 + ... + 32001 - (1 + 3 + 32 + 33 + ... + 32000)
2A = 3 + 32 + 33 + ... + 32001 - 1 - 3 - 32 - 33 - ... - 32000
2A = 32001 - 1
A = \(\dfrac{3^{2001}-1}{2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a) \(A=1+2+2^2+...+2^{50}\)
\(\Rightarrow2A=2+2^2+...+2^{51}\)
\(\Rightarrow A=2A-A=2+2^2+...+2^{51}-1-2-2^2-...-2^{50}=2^{51}-1\)
b) \(B=1+3+3^2+...+3^{100}\)
\(\Rightarrow3B=3+3^2+...+3^{101}\)
\(\Rightarrow2B=3B-B=3+3^2+...+3^{101}-1-3-3^2-...-3^{100}=3^{101}-1\)
\(\Rightarrow B=\dfrac{3^{101}-1}{2}\)
c) \(C=5+5^2+...+5^{30}\)
\(\Rightarrow5C=5^2+5^3+...+5^{31}\)
\(\Rightarrow4C=5C-C=5^2+5^3+...+5^{31}-5-5^2-...-5^{30}=5^{31}-5\)
\(\Rightarrow C=\dfrac{5^{31}-5}{4}\)
d) \(D=2^{100}-2^{99}+2^{98}-...+2^2-2\)
\(\Rightarrow2D=2^{101}-2^{100}+2^{99}-...+2^3-2^2\)
\(\Rightarrow3D=2D+D=2^{101}-2^{100}+2^{99}-...+2^3-2^2+2^{100}-2^{99}+...+2^2-2=2^{101}-2\)
\(\Rightarrow D=\dfrac{2^{101}-2}{3}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=2+2^2+...+2^{20}\)
\(2A=2^2+2^3+...+2^{21}\)
\(2A-A=2^2+2^3+...+2^{21}-2-2^2-...-2^{20}\)
\(A=2^{21}-2\)
___________
\(B=5+5^2+...+5^{50}\)
\(5B=5^2+5^3+...+5^{51}\)
\(5B-B=5^2+5^3+...+5^{51}-5-5^2-...-5^{50}\)
\(4B=5^{51}-5\)
\(B=\dfrac{5^{51}-5}{4}\)
___________
\(C=1+3+3^2+...+3^{100}\)
\(3C=3+3^2+...+3^{101}\)
\(3C-C=3+3^2+...+3^{101}-1-3-3^2-...-3^{100}\)
\(2C=3^{101}-1\)
\(C=\dfrac{3^{101}-1}{2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
`#3107`
\(A=1+2^1+2^2+2^3+...+2^{2015}\)
\(2A=2+2^2+2^3+2^4+...+2^{2016}\)
\(2A-A=\left(2+2^2+2^3+2^4+...+2^{2016}\right)-\left(1+2+2^2+2^3+...+2^{2015}\right)\)
\(A=2+2^2+2^3+2^4+...+2^{2016}-1-2-2^2-2^3-...-2^{2015}\)
\(A=2^{2016}-1\)
Vậy, \(A=2^{2016}-1.\)
\(A=2^0+2^1+2^2+...+2^{2015}\)
\(2\cdot A=2^1+2^2+2^3+...+2^{2016}\)
\(A=2A-A=2^{2016}-2^0\)
\(A=2^{2016}-1\)
![](https://rs.olm.vn/images/avt/0.png?1311)
A=1+2+22+23+...+262+263
2A=2+22+23+24+...+263+264
2A-A=2+22+23+24+...+263+264-1+2+22+23+...+262+263
A=264-1
\(C=1+2^2+2^3+...+2^{50}\)
\(3C=2+2^3+2^4+...+2^{51}\)
\(=>3C-C=2^{51}-1\)
\(=>C=\dfrac{2^{51}-1}{2}\)
\(#tnam\)