Light clock under constant proper accelleration

Jul 2008
Melbourne, Australia
I am consistently getting the wrong answer to this question and am starting to wonder if there is an error in the text.

A light clock has proper length l and moved longitudinally through an IF with proper accelleration alpha. (Ignore any variation of alpha along the rod). By looking at the time it takes the photon to make one to-and-fro bounce in the instantaneous rest frame, show that the frequency and proper frequency are related in the lowest order by

\(\displaystyle v=v_0 \gamma^{-1}(1+\frac 12 \alpha l/c^2)\)

the trouble is I get:
\(\displaystyle v=v_0 \gamma^{-1}(1+\frac 12 \alpha^2 l^2/c^4)\)

Which is correct? Any clues how to solve this?

Source: Rindler Relativity SG&C prob 3.10