Let's solve this problem step by step.

Problem:

Find three consecutive integers if the sum of the first two is 51 more than the third integer.

Step-by-Step Solution:

1. Define the integers: Let the three consecutive integers be:

   • First integer: $x$
   • Second integer: $x+1$
   • Third integer: $x+2$

2. Set up the equation: According to the problem, the sum of the first two integers is 51 more than the third integer. This can be written as: $(x) + (x+1) = (x+2) + 51$

3. Simplify the equation: $2x + 1 = x + 53$

4. Solve for $x$: Subtract $x$ from both sides: $x + 1 = 53$ Subtract 1 from both sides: $x = 52$

5. Find the integers:

   • First integer: $52$
   • Second integer: $52 + 1 = 53$
   • Third integer: $52 + 2 = 54$

Thus, the three consecutive integers are 52, 53, and 54.

Would you like a detailed explanation of any step? Here are some related questions you might find interesting:

1. What would the integers be if the sum of the first two were 60 more than the third?
2. How do we define consecutive odd or even integers algebraically?
3. Can we solve this problem by defining the integers differently?
4. What happens if we are asked for four consecutive integers with a similar condition?
5. How do we set up an equation if the problem involves the product instead of the sum?

Tip: Always start by defining variables clearly and setting up an equation based on the given conditions. This will help you solve the problem systematically.