Note that $n \equiv S(n) \bmod 9$, so $S(n+1)-S(n) \equiv n+1-n=1 \bmod 9$. So, since $S(n)=1274 \equiv 5 \bmod 9 $, we have that $S(n+1) \equiv 6 \bmod 9$. Then, only one of the answer choices is congruent to $6 \bmod 9$, which is $(D)=1239$. 
Every positive integer is congruent modulo $9$ to the sum of its decimal digits.  Now, let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507)=13$. For a particular positive integer $n, S(n)=1274$. Which of the following could be the value of $S(n+1)$? (Adapted From 2017 AMC 12A Problems, Question \#18) 

A. $$1$$ 
B. $$3$$ 
C. $$12$$ 
D. $$1239$$ 
E. $$1265$$ 