\(1.\\ a,\Rightarrow2S=2+2^2+2^3+...+2^{2023}\\ \Rightarrow2S-S=\left(2+2^2+...+2^{2023}\right)-\left(1+2+...+2^{2022}\right)\\ \Rightarrow S=2^{2023}-1\\ b,\Rightarrow3S=3^2+3^3+...+3^{2023}\\ \Rightarrow3S-S=\left(3^2+3^3+...+3^{2023}\right)-\left(3+3^2+...+3^{2022}\right)\\ \Rightarrow2S=3^{2023}-3\\ \Rightarrow S=\dfrac{3^{2023}-3}{2}\\ c,\Rightarrow4S=4^2+4^3+...+4^{2023}\\ \Rightarrow3S=4^{2023}-4\\ \Rightarrow S=\dfrac{4^{2023}-4}{3}\\ d,\Rightarrow5S=5^2+5^3+...+5^{2023}\\ \Rightarrow4S=5^{2023}-5\\ \Rightarrow S=\dfrac{5^{2023}-5}{4}\)
\(2,\\ A=1^2+2^2+...+20^2\\ A=1\left(2-1\right)+2\left(3-1\right)+...+20\left(21-1\right)\\ A=1\cdot2-1+2\cdot3-2+...+20\cdot21-20\\ A=\left(1\cdot2+2\cdot3+...+20\cdot21\right)-\left(1+2+...+20\right)\)
Đặt \(B=1\cdot2+2\cdot3+...+20\cdot21\)
\(\Rightarrow3B=1\cdot2\cdot3+2\cdot3\cdot3+...+20\cdot21\cdot3\\ \Rightarrow3B=1\cdot2\left(3-0\right)+2\cdot3\left(4-1\right)+...+20\cdot21\left(22-19\right)\\ \Rightarrow3B=1\cdot2\cdot3-1\cdot2\cdot3+2\cdot3\cdot4-...-19\cdot20\cdot21+20\cdot21\cdot22\\ \Rightarrow3B=20\cdot21\cdot22\\ \Rightarrow B=20\cdot7\cdot22=3080\)
\(\Rightarrow A=3080-\dfrac{\left(20+1\right)\left(20-1+1\right)}{2}=2870\)
Bài 3: b. (x - 5)2022 = (x - 5)2021
<=> 2022 = 2021 (Vô lí)
Vậy PT vô nghiệm
\(S=1+2+2^2+2^3+...+2^{2022}\)
\(2S=2+2^2+2^3+2^4+...+2^{2023}\)
\(2S-S=\left(2+2^2+2^3+2^4+...+2^{2023}\right)-\left(1+2+2^2+2^3+...+2^{2022}\right)\)
\(S=2^{2023}-1\)
