Question

A=3+3²+3³+...+3²⁰²¹

Tìm số dư khi tổng A chia cho 13.

Answer 1

Ta có A + 1 = 1 + 3 + 3² + 3³ + 3⁴ + 3⁵ + ... + 3²⁰¹⁹ + 3²⁰²⁰ + 3²⁰²¹ 

= (1 + 3 + 3²) + (3³ + 3⁴ + 3⁵) + ... + (3²⁰¹⁹ + 3²⁰²⁰ + 3²⁰²¹)

= (1 + 3 + 3²) + 3³(1 + 3 + 3²) + ... + 3²⁰¹⁹(1 + 3 + 3²)

= (1 + 3 + 3²)(1 + 3³ + ... +  3²⁰¹⁹)

= 13(1 + 3³ + ... +  3²⁰¹⁹) \(⋮\)13

=> A : 13 dư 12

Answer 2

\(A=3+3^2+3^3+...+3^{2021}\)

\(A=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^{2019}+3^{2020}+3^{2021}\right)\)

\(A=3\left(1+3+3^2\right)+3^4\left(1+3+3^2\right)+...+3^{2019}\left(1+3+3^2\right)\)

\(A=3.13+3^4.13+...+3^{2019}.13\)

\(A=13\left(3+3^4+...+3^{2019}\right)\)

\(\Rightarrow A⋮13\)

Hay \(A:13\)k dư