Let's define the three consecutive integers as:

• $x$ (the first integer),
• $x + 1$ (the second integer),
• $x + 2$ (the third integer).

According to the problem, the sum of the first and the third integer is 15 greater than the second integer. This can be written as:

$x + (x + 2) = (x + 1) + 15$

Simplifying the equation:

$2x + 2 = x + 16$

Subtract $x$ from both sides:

$x + 2 = 16$

Now, subtract 2 from both sides:

$x = 14$

So the three consecutive integers are $14$, $15$, and $16$.

Answer: 14, 15, 16

Would you like more details on how I solved this? Here are some related questions:

1. How would you solve for consecutive integers if the sum of the first and the third was something other than 15?
2. How can you generalize the process for any problem involving consecutive integers?
3. What is the formula to represent three consecutive integers algebraically?
4. How would the solution change if we were asked for four consecutive integers?
5. Can this method be applied to solve inequalities involving consecutive numbers?

Tip: Always translate word problems into algebraic expressions first. It makes the problem much clearer and easier to solve!