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
\(\ds \) \(=\) \(\ds \sum_{j \mathop = 0}^n 2^j \binom n j\)

$\blacksquare$