The difference between the squares of any two consecutive integers is equal to

### Concept & Formula This problem tests your knowledge of the algebraic expansion of consecutive integers and the difference of squares formula. $$ (x + y)(x - y) = x^2 - y^2 $$ ### Step-by-Step Solution * **Given:** We need to find the difference between the squares of two consecutive integers. * **Step 1: Define the variables.** Let the smaller integer be $n$. The consecutive integer directly following it will be $(n + 1)$. * **Step 2: Set up the equation.** We need to find $(n + 1)^2 - n^2$. * **Step 3: Expand the expression.** * $(n + 1)^2 - n^2 = (n^2 + 2n + 1) - n^2$ * $= 2n + 1$ * **Step 4: Interpret the result.** We can rewrite $2n + 1$ by breaking it apart into our original numbers: * $2n + 1 = n + (n + 1)$ * This perfectly matches the sum of our original two consecutive integers. ### Exam Strategy & Shortcut **Use real numbers:** Pick any two small, easy consecutive integers, like 3 and 4. Calculate the difference of their squares: $4^2 - 3^2 = 16 - 9 = 7$. Now, evaluate the options using the numbers 3 and 4: (a) Is 7 an even number? No. (b) Is 7 the difference of the given numbers ($4 - 3 = 1$)? No. (c) Is 7 the sum of the given numbers ($4 + 3 = 7$)? Yes! ### Common Pitfall Students often try to memorize this as a standalone rule rather than recognizing it as a simple application of $a^2 - b^2$. Because $(a-b)$ for consecutive integers is always 1, $a^2 - b^2$ simplifies immediately to $1 \times (a+b)$, which is just the sum! ### Final Answer Therefore, the correct answer is **sum of given numbers**.