On the difference between consecutive primes

Update: This work reproduces an earlier result of Peck, which the author was
initially unaware of. The method of the proof is essentially the same as the
original work of Peck. There are no new results.
We show that the sum of squares of differences between consecutive primes
$\sum_{p_n\le x}(p_{n+1}-p_n)^2$ is bounded by $x^{5/4+{\epsilon}}$ for $x$
sufficiently large and any fixed ${\epsilon}>0$. The proof relies on utilising
various mean-value estimates for Dirichlet polynomials.