The right answer will be W. Each alphabet here is repeated the number of times of its positional sequence in the series of alphabets. 1 A , 2 Bs, 3 Cs and so on. So we want to know where 288 comes as a result in the sequence 1+2+3+.......... and so on. This sequence goes in an arithmetic progression with a sum characteristics with a formula x=n*(n+1)/2 or 288=n*(n+1)/2 solving for n we get something that starts with 23 and ends with 24. W is a positional alphabet for 23 that will be start at a position 277 and will be repeated 23 times before the start of X that will start at a position of 301. So, the 288 th position will obviously be W and not X.
If you will solve in detail... you will come to find out it is X.... It is a CAT question and the answer was X.... try solving once more.... I did solve first with the natural progression method.,.. and then I sat to check it down... and I got X