Breaking a Modified Caesar Cipher
Assume the length is $n$. If the cipher text is $c_0, c_1, c_2, \ldots, c_N$ then consider the sub-text consisting of the characters $c_0, c_n, c_{2n},\ldots$. These have all been encrypted with the same Caesar, and you can break it by frequency analysis (the shifted 'e' should be the most common in standard English texts, or else maybe the 't' etc.). Suppose the shift you find is $s_0$. Then try the shifts $s_0 + 1, \ldots s_0 + (n-1)$ for the next sub-texts and see if you have a result. If not, try another $n$. This way you only brute force over $n$, essentially.

You could also determine the length $n$ by a statistical method (index of coincidence) or Kasiski (finding long repeats) as well. These will work for any Viginère cipher. But the above might be simpler if you already know that the shifts will be cyclic and consecutive.