\(B>\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{30.31}\)

\(B>\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{30}-\dfrac{1}{31}\)

\(B>\dfrac{1}{2}-\dfrac{1}{31}=\dfrac{29}{62}\left(đpcm\right)\)

\(S=2^1+2^2+2^3+2^4+2^5+2^6+..+2^{28}+2^{29}+2^{30}\) 

\(S=2.\left(1+2+2^2\right)+2^4.\left(1+2+2^2\right)+...+2^{28}.\left(1+2+2^2\right)\) 

\(S=\left(1+2+2^2\right).\left(2+2^4+...+2^{28}\right)\) 

\(S=7.\left(2+2^4+...+2^{28}\right)\) 

⇒ \(S⋮7\)   ( điều phải chứng minh ) 

S=2¹+2²+2³+...+2³⁰

S=(2¹+2²+2³)+(2⁴+2⁵+2⁶)+...+(2²⁸+2²⁹+2³⁰)

S=7.2+7.2⁴+...+7.2²⁸

S=7.(2+2⁴+...+2²⁸)

⇒S⋮7

Ta có: \(S=2^1+2^2+2^3+...+2^{28}+2^{29}+2^{30}\)

\(=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+...+2^{28}\left(1+2+2^2\right)\)

\(=7\cdot\left(2+2^4+...+2^{28}\right)⋮7\)

Ta có : A = 1 + 2 + 2² + ..... + 2³⁰ 

=> 2A = 2 + 2² + ..... + 2³¹

=> 2A - A = 2³¹ - 1

=> A = 2³¹ - 1 (đpcm)

\(A=1+2+2^2+.......+2^{29}+2^{30}\)

\(2A=2.\left(1+2+2^2+........+2^{29}+2^{30}\right)\)

\(2A=2+2^2+2^3+......+2^{30}+2^{31}\)

\(2A-A=\left(2+2^2+2^3+.....+2^{30}+2^{31}\right)-\left(1+2+2^2.....+2^{29}+2^{30}\right)\)

\(A=2^{31}-1\)

\(\Rightarrow A=1+2+2^2+......+2^{29}+2^{30}=2^{31}-1\)

TA CÓ:

A=3⁰+3+3²+3³+........+3¹¹

(3⁰+3+3²+3³)+....+(3⁸+3⁹+3¹⁰+3¹¹)

3(0+1+3+3²)+......+3⁸(0+1+3+3²) 

3.13+....+3⁸.13 cHIA HẾT CHO 13 NÊN A CHIA HẾT CHO 13( đpcm)