Câu hỏi của Trần Đại Nghĩa

Question

Tính:$S=3-3+6-9+12+24-27+48-81+...-2187+3072+6144-6561+12288$

Nguyễn Linh Chi · Accepted Answer

Ta có: $S=3.2^0-3^1+3.2^1-3^2+3.2^2+3.2^3-3^3+3.2^4-3^4+...-3^7+3.2^{10}+3.2^{11}-3^8+3.2^{12}$$=3.\left(2^0+2^1+2^2+2^3+2^4+...+2^{10}+2^{11}+2^{12}\right)-\left(3^1+3^2+3^3+...+3^7+3^8\right)$Đặt: $A=2^0+2^1+2^2+...+2^{11}+2^{12}$=> $2.A=2^1+2^2+2^3+...+2^{12}+2^{13}$=> $2.A-A=2^{13}-2^0$$\Rightarrow A=2^{13}-1=8191$Đặt: $B=3^1+3^2+3^3+...+3^8$ $\Rightarrow3.B=3^2+3^3+3^4+...+3^9$=> $3B-B=3^9-3^1=19680$=> $2B=19680\Rightarrow B=9840$=> S=3.A-B=3.8191-9840=14733Câu trả lời: Ta có: $S=3.2^0-3^1+3.2^1-3^2+3.2^2+3.2^3-3^3+3.2^4-3^4+...-3^7+3.2^{10}+3.2^{11}-3^8+3.2^{12}$$=3.\left(2^0+2^1+2^2+2^3+2^4+...+2^{10}+2^{11}+2^{12}\right)-\left(3^1+3^2+3^3+...+3^7+3^8\right)$Đặt: $A=2^0+2^1+2^2+...+2^{11}+2^{12}$=> $2.A=2^1+2^2+2^3+...+2^{12}+2^{13}$=> $2.A-A=2^{13}-2^0$$\Rightarrow A=2^{13}-1=8191$Đặt: $B=3^1+3^2+3^3+...+3^8$ $\Rightarrow3.B=3^2+3^3+3^4+...+3^9$=> $3B-B=3^9-3^1=19680$=> $2B=19680\Rightarrow B=9840$=> S=3.A-B=3.8191-9840=14733

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