Question

Tìm số tự nhiên n thoả mãn 2.2^2+3.2^3+4.2^4+...+n.2^n=2^n+11

Answer 1

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Answer 2

Đặt A=2.22+3.23+4.24+...+n.2nA=2.22+3.23+4.24+...+n.2n

Ta có:

A=2.22+3.23+4.24+...+n.2nA=2.22+3.23+4.24+...+n.2n

⇒2A=2(2.22+3.23+4.24+...+n.2n)⇒2A=2(2.22+3.23+4.24+...+n.2n)

⇒2A=2.23+3.24+4.25+...+n.2n+1⇒2A=2.23+3.24+4.25+...+n.2n+1

⇒2A−A=2.22+(3.23−2.23)+...+(n−n+1).2n−n.2n+1⇒2A−A=2.22+(3.23−2.23)+...+(n−n+1).2n−n.2n+1

⇒A=2.22+23+24+...+2n−n.2n+1⇒A=2.22+23+24+...+2n−n.2n+1

⇒A=22+(22+23+...+2n+1)−(n+1).2n+1⇒A=22+(22+23+...+2n+1)−(n+1).2n+1

⇒A=−22−(22+23+...+2n+1)+(n+1).2n+1⇒A=−22−(22+23+...+2n+1)+(n+1).2n+1

Đặt B=22+23+...+2n+1B=22+23+...+2n+1

⇒2B=23+24+...+2n+2⇒2B=23+24+...+2n+2

⇒2B−B=2n+2−22⇒B=2n+2−22⇒2B−B=2n+2−22⇒B=2n+2−22

⇒A=22−2n+2+22+(n+1).2n+1⇒A=22−2n+2+22+(n+1).2n+1

⇒A=(n+1).2n+1−2n+2⇒A=(n+1).2n+1−2n+2

⇒A=2n+1(n+1−2)⇒A=2n+1(n+1−2)

⇒A=(n−1).2n+1=2(n−1).2n⇒A=(n−1).2n+1=2(n−1).2n

Mà A=2(n−1).2n=2n+10A=2(n−1).2n=2n+10

⇒2(n+1)=210⇒n−1=29⇒2(n+1)=210⇒n−1=29

⇒n−1=512⇒n=513⇒n−1=512⇒n=513

Vậy n=513