\(3B=1.3^2+2.3^3+3.3^4+...+2022.3^{2023}+2023.3^{2024}\)
\(2B=3B-B=-3-3^2-3^3-...-3^{2023}+2023.3^{2024}\)
\(2B=2023.3^{2024}-\left(3+3^2+3^3+...+3^{2023}\right)\)
Đặt
\(C=3+3^2+3^3+...+3^{2023}\)
\(3C=3^2+3^3+3^4+...+3^{2024}\)
\(2C=3C-C=3^{2024}-3\Rightarrow C=\dfrac{3^{2024}-3}{2}\)
\(\Rightarrow2B=2023.3^{2024}-\dfrac{3^{2024}-3}{2}=\)
\(=\dfrac{2.2023.3^{2024}-3^{2024}+3}{2}=\dfrac{4045.3^{2024}+3}{2}\)
\(\Rightarrow B=\dfrac{4045.3^{2024}+3}{4}\)
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