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Compute the unique positive integer $n$ such that \(2 \cdot 2^2 + 3 \cdot 2^3 + 4 \cdot 2^4 + \dots + n \cdot 2^n = 2^{n + 10}.\)

 Feb 13, 2020
 #1
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By arithmetico-geometric series, the series on the left is 2^n*(n - 2).  Then n - 2 = 2^10 = 1024, so n = 1026.

 Feb 13, 2020
 #2
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sumfor(n,2, 17,n*2^n =4194304 =2^(12 + 10)

 Feb 13, 2020
 #3
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Forget  answer #2 !!.

Guest Feb 13, 2020
 #4
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a=2;c=0; b=a * 2^a;c=c+b; if(c==2^(a+10), goto5, goto6);printc, a, b; a++;if(a<2000, goto2, 0)

 

OUTPUT = n = 513.               2. 2^2 + 3. 2^3 + 4. 2^4 +............+ 513 * 2^513 = 2^(513 +10)

 Feb 13, 2020
 #5
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+5

I agree with guest #4:

 

 Feb 13, 2020

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