Explanation:

When three integers are in arithmetic progression, the average of the numbers will be equal to the middle number. Assuming this middle number as a, we can express the three numbers as,
(a−x), a and (a+x), where common difference between two adjacent numbers is x.
Adding,
3a=30,
Or, a=10.
Multiplying together,
10(100−x²)=910,
Or, 100−x²=91,
Or, x=3.
The largest number is,
a+x=13.