Consecutive integers — The difference between the squares of two consecutive numbers is 35. Identify the two numbers.

Introduction / Context:For consecutive integers n and n+1, the difference of their squares has a well-known identity that makes the calculation immediate. Recognizing this identity avoids unnecessary computation and speeds up problem solving.

Given Data / Assumptions:

• Two numbers are consecutive: n and n + 1.
• Given: (n + 1)^2 - n^2 = 35.
• We solve for integer n.

Concept / Approach:Use the difference of squares identity: (a^2 - b^2) = (a - b) * (a + b). For a = n + 1 and b = n, we get (n + 1)^2 - n^2 = (1) * (2n + 1) = 2n + 1. Set 2n + 1 equal to 35 and solve.

Step-by-Step Solution:Write identity: (n + 1)^2 - n^2 = 2n + 1.Set equal to 35: 2n + 1 = 35.Solve: 2n = 34 → n = 17.Therefore, the numbers are 17 and 18.

Verification / Alternative check:Compute directly: 18^2 - 17^2 = 324 - 289 = 35, exactly as required. This confirms the identity-based solution.

Why Other Options Are Wrong:

• 14,15 → 225 - 196 = 29; 15,16 → 256 - 225 = 31; 18,19 → 361 - 324 = 37; 16,17 → 289 - 256 = 33. None equals 35.

Common Pitfalls:Squaring both numbers and subtracting without using the identity (slower and error-prone); forgetting that for consecutive numbers, the difference of squares simplifies to 2n + 1, an odd number.

Final Answer:17, 18