What is the sum of two consecutive odd numbers, the difference of whose…

Concept: For any two numbers that differ by a fixed amount, the difference of their squares can be found directly from the identity a2 − b2 = (a − b)(a + b) — it equals the product of their difference and their sum. Since two consecutive odd numbers always differ by 2, this identity turns the given condition on their squared difference into a simple linear equation in one unknown.

Application:

1. Let the smaller consecutive odd number be x; then the next is x + 2.

2. The difference of their squares is (x + 2)2 − x2 = 4x + 4.

3. Set 4x + 4 = 56 and solve: 4(x + 1) = 56 ⇒ x + 1 = 14 ⇒ x = 13.

4. Thus the numbers are 13 and 15, and their sum is 13 + 15 = 28.

Verification: 152 − 132 = 225 − 169 = 56, so the solution is consistent.