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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 = 32.

 Jan 10, 2024
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\(2 \cdot 2^2 + 3 \cdot 2^3 + 4 \cdot 2^4 + \dots + n \cdot 2^n = 32.\)

 

We can calculate that \(2 \cdot 2^2 \)is 8, and that \( 3 \cdot 2^3 \) is 24, and because 8 + 24 is 32, then n is 3, and the equation \(2 \cdot 2^2 + 3 \cdot 2^3 + 4 \cdot 2^4 + \dots + n \cdot 2^n = 32\) is \(2 \cdot 2^2 + 3 \cdot 2^3 =32\)

 

Answer: 3

 Jan 10, 2024

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