Sum of Sequence of Binomial Coefficients by Powers of 2/Proof 1
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Theorem
\(\ds \sum_{j \mathop = 0}^n 2^j \binom n j\) | \(=\) | \(\ds \dbinom n 0 + 2 \dbinom n 1 + 2^2 \dbinom n 2 + \dotsb + 2^n \dbinom n n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3^n\) |
Proof
\(\ds 3^n\) | \(=\) | \(\ds \paren {2 + 1}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 0}^n 2^j 1^{n - j} \binom n j\) | Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 0}^n 2^j \binom n j\) |
$\blacksquare$