Question

[Nguyễn Huy Hải]

Tìm n thỏa mãn 2.2² + 3.2³ + 4.2⁴ + 5.2⁵ + ... + n.2ⁿ = 2^{n + 10}

Answer 1

[Lê Chí Cường]

Đặt \(A=2.2^2+3.2^3+4.2^4+...+n.2^n\)

=>\(2.A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}\)

=>\(A-2A=2.2^2+3.2^3+4.2^4+...+n.2^n-2.2^3-3.2^4-4.2^5-...-n.2^{n+1}\)

=>\(-A=2.2^2+\left(3.2^3-2.2^3\right)+\left(4.2^4-3.2^4\right)+...+\left(n.2^n-\left(n-1\right).2^n\right)-n.2^{n+1}\)

=>\(-A=2^3+2^3+2^4+...+2^n-n.2^{n+1}\)

=>\(-A=2^3+\left(2^3+2^4+...+2^n\right)-n.2^{n+1}\)

=>\(A=n.2^{n+1}-2^3-\left(2^3+2^4+...+2^n\right)\)

Đặt \(B=2^3+2^4+...+2^n\)

=>\(2.B=2^4+2^5+...+2^{n+1}\)

=>\(2.B-B=2^4+2^5+...+2^{n+1}-2^3-2^4-...-2^n\)

=>\(B=2^{n+1}-2^3\)

Lại có:\(A=n.2^{n+1}-2^3-\left(2^3+2^4+...+2^n\right)\)

=>\(A=n.2^{n+1}-2^3-B\)

=>\(A=n.2^{n+1}-2^3-\left(2^{n+1}-2^3\right)\)

=>\(A=n.2^{n+1}-2^3-2^{n+1}+2^3\)

=>\(A=n.2^{n+1}-2^{n+1}\)

=>\(A=\left(n-1\right).2^{n+1}\)

Mà \(A=2.2^2+3.2^3+4.2^4+...+n.2^n=2^{n+10}\)

=>\(\left(n-1\right).2^{n+1}=2^{n+10}\)

=>\(n-1=2^{n+10}:2^{n+1}\)

=>\(n-1=2^{n+10-n-1}\)

=>\(n-1=2^9\)

=>\(n-1=512\)

=>\(n=513\)

Vậy n=513

Answer 2

[YêU xÔ đẤy Có SaO kHôNg]

dài thế hình như cô giáo lớp mình giải còn ngắn hơn thế này