Tham khảo : Tìm số tự nhiên n thoả mãn 2.2^2 +3.2^3 +4.2^4+ ...+n.2^n = 2^n+11 - My Hien

Tham khảo : Tìm số tự nhiên n thoả mãn - My Hien

https://h.vn/hoi-dap/question/221389.html kham khảo ak!!! bài dài quá lười đánh máy lắm, thông cảm!!!^~

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

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

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

S=2.2^2+3.2^3+...+n.2^n=2^{n+11}

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

S=n.2^{n+1}-2^3-(2^3+2^4+...+2^{n-1}+2^n)

Dat T=2^3+2^4+...+2^{n-1}+2^n

Ta tinh dc: T=2T-T=2^{n-1}-2^3

S=n.2^{n+1}-2^3-2^{n-1}+2^3=(n-1).2^{n+1}

=> (n-1).2^{n+1}=n^{n+11}

=> n-1=2^{10}

=> n=2^{10}+1

=> n=1024+1

=> n = 1025

trả lời 

n=1025 

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