Select the correct answerThe difference between the squares of two consecutive numbers is 35. The numbers are?Options17&1819&2016&1818&1717&20

1. Break Down the Problem

We need to find two consecutive integers, $n$ and $n+1$, such that the difference between their squares is 35: $(n+1)^2 - n^2 = 35$

2. Relevant Concepts

Using the difference of squares formula, we can simplify: $(n+1)^2 - n^2 = (n^2 + 2n + 1) - n^2 = 2n + 1$

3. Analysis and Detail

Set the equation from the difference of squares equal to 35: $2n + 1 = 35$ Now solve for $n$: $2n = 35 - 1 \\ 2n = 34 \\ n = 17$ Thus, the two consecutive numbers are: $n = 17 \quad \text{and} \quad n+1 = 18$

4. Verify and Summarize

Verify the solution by checking the difference of the squares: $18^2 - 17^2 = 324 - 289 = 35$ Since the calculation is correct, we conclude that the numbers are indeed consecutive.

Final Answer

The numbers are $17$ and $18$.