Question

Tổng sau có chia hết cho 3 không? Giải thích

A = 2 + 2² + 2³ + 2⁴ + 2⁵ + 2⁶ + 2⁷ + 2⁸ + 2⁹ + 2¹⁰

Answer 1

Ta có: \(A=2+2^2+2^3+2^4+...+2^{10}\)

\(\Rightarrow A=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^9+2^{10}\right)\)

\(\Rightarrow A=2.\left(1+2\right)+2^3.\left(1+2\right)+....+2^9.\left(1+2\right)\)

\(\Rightarrow A=3.\left(2+2^3+...+2^9\right)⋮3\)

Vậy A chia hết cho 3

Answer 2

A = 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^9 + 2^10

= ( 2 + 2^2 ) + ( 2^3 + 2^4 ) + ( 2^5 + 2^6 ) + ( 2^7 + 2^8 ) + ( 2^9 + 2^10 )

= 2 x ( 1 + 2 ) + 2^3 x ( 1 + 2 ) + 2^5 x ( 1 + 2 ) +2^7 x ( 1 + 2 ) + 2^9 x ( 1 + 2 )

= 2 x 3 + 2^3 x 3 + 2^5 x 3 + 2^7 x 3 + 2^9 x 3

= 3 x ( 2 + 2^3 + 2^5 + 2^7 + 2^9) chia hết cho 3 

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