Can anyone help prove this binomial identity by using some elementary methods, and I have no clue to get started. $$\sum_{k=0}^{n}\binom{n+k}{k} \frac{1}{2^k}=2^n$$ Wolfram Mathematica can transform the function $\sum_{k=0}^{n}\binom{n+k}{k} x^k$ to something contains $\Gamma$ function and incomplete Beta-function, by using hypergeometric series, but it's not a simple method anymore.
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1There is an error in your equation, the right-hand side should be $2^n$. – Martin R Sep 17 '21 at 09:06
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Another one: https://math.stackexchange.com/q/3570373/42969 – both found with Approach0 – Martin R Sep 17 '21 at 09:06