Câu hỏi của ❖ Khang/GD❄ 『ʈєɑɱ❖Hoàng๖ۣۜGia☆』

Question

a, A = 1 + 2 + 22 + 23 + ... + 250 =b, B = 1 + 3 + 32 + 33 + ... 3100 =c, C = 5 + 52 + 53 + ... 530 = d, D = 2100 = 299 + 298 - 297 + ... + 22 - 2

Lấp La Lấp Lánh · Accepted Answer

a) $A=1+2+2^2+...+2^{50}$$\Rightarrow2A=2+2^2+...+2^{51}$$\Rightarrow A=2A-A=2+2^2+...+2^{51}-1-2-2^2-...-2^{50}=2^{51}-1$b) $B=1+3+3^2+...+3^{100}$$\Rightarrow3B=3+3^2+...+3^{101}$$\Rightarrow2B=3B-B=3+3^2+...+3^{101}-1-3-3^2-...-3^{100}=3^{101}-1$$\Rightarrow B=\dfrac{3^{101}-1}{2}$c) $C=5+5^2+...+5^{30}$$\Rightarrow5C=5^2+5^3+...+5^{31}$$\Rightarrow4C=5C-C=5^2+5^3+...+5^{31}-5-5^2-...-5^{30}=5^{31}-5$$\Rightarrow C=\dfrac{5^{31}-5}{4}$d) $D=2^{100}-2^{99}+2^{98}-...+2^2-2$$\Rightarrow2D=2^{101}-2^{100}+2^{99}-...+2^3-2^2$$\Rightarrow3D=2D+D=2^{101}-2^{100}+2^{99}-...+2^3-2^2+2^{100}-2^{99}+...+2^2-2=2^{101}-2$$\Rightarrow D=\dfrac{2^{101}-2}{3}$Câu trả lời: a) $A=1+2+2^2+...+2^{50}$$\Rightarrow2A=2+2^2+...+2^{51}$$\Rightarrow A=2A-A=2+2^2+...+2^{51}-1-2-2^2-...-2^{50}=2^{51}-1$b) $B=1+3+3^2+...+3^{100}$$\Rightarrow3B=3+3^2+...+3^{101}$$\Rightarrow2B=3B-B=3+3^2+...+3^{101}-1-3-3^2-...-3^{100}=3^{101}-1$$\Rightarrow B=\dfrac{3^{101}-1}{2}$c) $C=5+5^2+...+5^{30}$$\Rightarrow5C=5^2+5^3+...+5^{31}$$\Rightarrow4C=5C-C=5^2+5^3+...+5^{31}-5-5^2-...-5^{30}=5^{31}-5$$\Rightarrow C=\dfrac{5^{31}-5}{4}$d) $D=2^{100}-2^{99}+2^{98}-...+2^2-2$$\Rightarrow2D=2^{101}-2^{100}+2^{99}-...+2^3-2^2$$\Rightarrow3D=2D+D=2^{101}-2^{100}+2^{99}-...+2^3-2^2+2^{100}-2^{99}+...+2^2-2=2^{101}-2$$\Rightarrow D=\dfrac{2^{101}-2}{3}$

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