The difference between the squares of two consecutive numbers is 35. The numbers are?Options19&2018&1716&1817&2017&18

1. Break Down the Problem

Let the two consecutive numbers be $n$ and $n + 1$. The difference between their squares can be expressed as: $(n + 1)^2 - n^2 = 35$

2. Relevant Concepts

Using the algebraic identity, we expand the difference of squares: $(n + 1)^2 - n^2 = (n^2 + 2n + 1) - n^2 = 2n + 1$ So now we have: $2n + 1 = 35$

3. Analysis and Detail

To solve for $n$, we can rearrange the equation: $2n = 35 - 1 \\ 2n = 34 \\ n = 17$ Thus, the two consecutive numbers are $n = 17$ and $n + 1 = 18$.

4. Verify and Summarize

We can verify our result by substituting back into the expression for the difference of squares: $18^2 - 17^2 = 324 - 289 = 35$ The calculations confirm that the difference is indeed 35.

Final Answer

The two consecutive numbers are 17 and 18.