Question

Cho A = 75.(4²⁰²³ + 4²⁰²⁴+ ... + 4² + 5) + 25. Chứng minh A ⋮ 4²⁰²⁴

Answer 1

Sửa đề: \(A=75\left(4^{2023}+4^{2022}+...+4^2+5\right)+25\)

Đặt \(B=4^{2023}+4^{2022}+...+4^2+5\)

=>\(B=4^{2023}+4^{2022}+...+4^2+4+1\)

=>\(4B=4^{2024}+4^{2023}+...+4^3+4^2+4\)

=>\(4B-B=4^{2024}+4^{2023}+...+4^3+4^2+4-4^{2023}-4^{2022}-...-4^2-4-1\)

=>\(3B=4^{2024}-1\)

=>\(B=\dfrac{4^{2024}-1}{3}\)

\(A=75\cdot B+25\)

\(=25\left(4^{2024}-1\right)+25=25\cdot4^{2024}⋮4^{2024}\)