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 is equivalent to the nth expression in the pattern.

The given pattern consists of expressions that can be simplified. Each expression can be rewritten as the difference of squares, leading to a consistent formula that generates the terms based on the index n.

A (2n - 1) CORRECT

This expression correctly represents the nth term of the pattern. Each term is derived from the difference of squares, specifically \(n^2 - (n-1)^2\), which simplifies to \(2n - 1\). This matches the incremental pattern observed in the initial terms provided.

B (2n^2 + 1) YOUR ANSWER

This expression does not match the derived formula from the pattern. While it includes the variable n, it suggests a quadratic relationship that does not align with the linear growth of the actual terms in the sequence. The correct approach shows a linear relationship rather than a quadratic one.

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

This choice also does not reflect the correct pattern. It suggests a different form involving quadratic and linear components without capturing the essence of the difference of squares. The derived expression from the pattern does not incorporate a negative term or constant like -1, which misrepresents the sequence.

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

This expression introduces unnecessary additional terms, suggesting a quadratic nature that does not represent the linear differences observed in the pattern. The correct nth term should directly relate to the linear progression of the pattern, which this option fails to achieve.

Conclusion

In summary, the nth expression in the pattern of \(1^2 - 0^2, 2^2 - 1^2, 3^2 - 2^2, ...\) simplifies to \(2n - 1\), accurately reflecting the linear relationship between n and the terms generated. The other choices either introduce incorrect polynomial forms or fail to capture the essence of the pattern's growth. Hence, 2n - 1 is the valid expression for the nth term.

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 is equivalent to the nth expression in the pattern.

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