31. 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, ...
Answer: A
2n - 1
The nth expression in the pattern corresponds to the formula 2n - 1, which can be derived from analyzing the differences between consecutive squares in the given sequence.
A) 2n - 1
This option is correct because the expressions follow the pattern of differences between consecutive squares: n^2 - (n-1)^2 simplifies to 2n - 1. Each term represents the difference between the square of n and the square of the previous integer, which matches the provided expressions.
B) 2n^2 + 1
This option is incorrect as it suggests a quadratic expression that does not represent the difference of squares. The pattern shows linear growth, while 2n^2 + 1 implies a parabolic increase, which does not fit the established sequence.
C) n^2 - n - 1
This option is also incorrect. While it is a quadratic expression, it does not correlate with the pattern observed in the differences of squares. The expression n^2 - n - 1 does not simplify to the difference of two squares, thus failing to represent the nth term accurately.
D) n^2 + n + 1
This option is incorrect as it suggests a different pattern entirely. The expression n^2 + n + 1 indicates a quadratic growth which does not align with the linear progression seen in the provided expressions. It does not represent the difference of squares or capture the essence of the pattern.
Conclusion
The correct answer, 2n - 1, directly reflects the difference between the squares of consecutive integers, accurately representing the nth term of the sequence. All other options fail to capture this relationship and do not align with the established pattern, confirming that they are not suitable representations of the nth expression.