Question

chứn tỏ rằng: 2+2^2+2^3+2^4+....+2^59+2^60+5^2021 chia hết cho 5cần gấp ặ

Answer 1

A = 2 + 2² + 2³ + 2⁴ + ... + 2⁵⁹ + 2⁶⁰ + 5²⁰²¹

= (2 + 2³) + (2² + 2⁴) + (2⁵ + 2⁷) +... + (2⁵⁷ + 2⁵⁹) + (2⁵⁸ + 2⁶⁰) + 5²⁰²¹ 

= 2(1 + 2²) + 2²(1 + 2²) + 2⁵(1 + 2²) + ... + 2⁵⁷(1 + 2²) + 2⁵⁸(1 + 2²) + 5²⁰²¹

= (1 + 2²)(2 + 2² + 2⁵ + ... + 2⁵⁷ + 2⁵⁸) + 5²⁰²¹ 

= 5(2 + 2² + 2⁵ + ... + 2⁵⁷ + 2⁵⁸) + 5²⁰²¹ 

= 5(2 + 2² + 2⁵ + ... + 2⁵⁷ + 2⁵⁸ + 5²⁰²⁰) \(⋮\)5

Answer 2

\(2+2^2+2^3+2^4+...+2^{59}+2^{60}+5^{2021}\)

\(=\left(2+2^3\right)+\left(2^2+2^4+\right)+\left(2^5+2^7\right)+...+\left(2^{57}+2^{59}\right)+\left(2^{58}+2^{60}\right)+5^{2021}\)

\(=2\left(1+2^2\right)+2^2\left(1+2^2\right)+2^5\left(1+2^2\right)+...+2^{57}\left(1+2^2\right)+2^{58}+\left(1+2^2\right)+5^{2021}\)\(=\left(1+1^2\right)\left(2+2^2+2^5+...2^{57}+2^{58}\right)+5^{2021}\)

\(=5\left(2+2^2+2^5+...+2^{57}+2^{58}\right)+5^{2021}\)

\(=5\left(2+2^2+2^5+...+2^{57}+2^{58}+5^{2021}\right)⋮5\)

\(\text{Hok tốt!}\)

\(\text{@Kaito Kid}\)