2.2^2+3.2^3+4.2^4+…+(n-1).2^n-1+n.2^n=2^n+34 tìm n

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1. Đáp án:

   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

   Giải thích các bước giải:

    10 điểm

   Trả lời

2. Giải thích các bước giải:

   Đặt `A=2.2^2+3.2^3+4.2^4+…+(n-1).2^(n-1)+n.2^n`

   `=> 2A = 2.2^3 + 3.2^4 + 4.2^5 + … + (n-1).2^n + n.2^(n+1)`

   `=> 2A-A= n . 2^(n+1) – 2^3 – (2^3 + 2^4 + …. + 2^(n-1) + 2^n)`

   Đặt `B = 2^3 + 2^4 + …. + 2^(n-1) + 2^n`

   `=> 2B = 2^4 + 2^5 + … + 2^n + 2^(n+1)`

   `=> 2B – B = (2^4 + 2^5 + … + 2^n + 2^(n+1)) – (2^3 + 2^4 + …. + 2^(n-1) + 2^n)`

   `=> B = 2^(n+1) – 2^3`

   `=> A = n.2^(n+1) – 2^3 – 2^3 – (2^(n+1) – 2^3)`

   `= n.2^(n+1) – 2^3 – 2^(n+1) + 2^3`

   `= n.2^(n+1) – 2^(n+1) = 2^(n+1) . (n-1)`

   `=> 2^(n+1) ( n-1) = 2^(n+34)`

   `=> n-1= 2^(n+34) : 2^(n+1)`

   `=> n-1 = 2^33`

   `=> n = 2^33 + 1`

   Trả lời