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Yes, this is true. In 1952, Nagura proved that for $n \geq 25$, there is always a prime between $n$ and $\frac{6}{5} n$. Thus, let $p_k$ be a prime at least $25$. Then $p_k+p_{k+1} > 2p_k$. But by Nagura's result we have that $p_{k+2} \leq \frac{36}{25} p_k < 2p_k$. Finally, one can easily check by hand that the result holds for small primes.