The first three expressions in a pattern are shown. Which of the following is equivalent to the nth expression in the pattern? 1^2 - 0^2, 2^2 - 1^2, 3^2 - 2^2, ...

Rationale

2n - 1

The nth expression in the pattern can be derived by recognizing the structure of the given expressions. Each expression is of the form n^2 - (n-1)^2, which simplifies to 2n - 1. This clearly shows that the pattern follows a linear expression based on n.

A (2n - 1) CORRECT

This option correctly represents the nth expression. When simplified, n^2 - (n-1)^2 becomes n^2 - (n^2 - 2n + 1), which simplifies to 2n - 1. Therefore, this expression accurately reflects the pattern established by the first three expressions.

B (2n^2 + 1) YOUR ANSWER

This option incorrectly suggests that the nth expression is quadratic in nature. If we analyze the pattern, the difference between consecutive squares does not yield a term with n^2; rather, it consistently results in a linear term. Thus, this option does not match the derived expression.

C (n^2 - n - 1) YOUR ANSWER

This choice also fails to conform to the pattern observed. When expanded, n^2 - n - 1 does not align with the simplification of the expressions provided. Each term in the sequence has a linear relationship with n, rather than a quadratic or mixed form as suggested here.

D (n^2 + n + 1) YOUR ANSWER

This option proposes a completely different structure from the observed pattern. It adds additional terms that do not exist in the original expressions. The consistent pattern of differences derived from the squares leads to a linear function, which this option does not represent.

Conclusion

The correct expression for the nth term in the pattern is 2n - 1, derived from the simplification of the given sequence. Each option was evaluated against the established pattern, revealing that only option A accurately reflects the linear relationship inherent in the expressions. The pattern showcases a fundamental principle of deriving differences between squares, leading to a consistently linear formula as n progresses.

The first three expressions in a pattern are shown. Which of the following is equivalent to the nth expression in the pattern? 1^2 - 0^2, 2^2 - 1^2, 3^2 - 2^2, ...

2n - 1

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