Question

Chứng minh rằng tổng A=1+2+2²+2³+2⁴+...+2²⁹+2³⁰ bằng( 2³¹ - 1)

 

Answer 1

Ta có : A = 1 + 2 + 2² + ..... + 2³⁰ 

=> 2A = 2 + 2² + ..... + 2³¹

=> 2A - A = 2³¹ - 1

=> A = 2³¹ - 1 (đpcm)

Answer 2

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

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

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

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

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

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