Each of the figures shown, except for n = 1, is composed of one or more small, identical equilateral triangles. If this pattern continues, which of the following expressions would represent the number of these equilateral triangles in a figure when n = k?

Rationale

(k - 1)^2

The expression (k - 1)^2 accurately represents the number of equilateral triangles in the figure when n = k, as it reflects the pattern established by the arrangement of triangles in the previous figures.

A ((k - 1)^2) CORRECT

This expression correctly captures the relationship between n and the number of equilateral triangles in the figure. For any value of n, the number of triangles is obtained by squaring (n - 1), reflecting the pattern where each increment in n results in a quadratic growth of the triangle count.

B (k^2) YOUR ANSWER

The expression k^2 suggests that the number of triangles increases quadratically with k, starting from k = 1. However, the pattern established shows that when n = 1, the figure does not contain any triangles, making this expression incorrect for the given pattern.

C ((k + 1)^2) YOUR ANSWER

This expression implies that the number of triangles increases with an initial offset of one triangle more than expected. This does not align with the observed pattern, as the initial figure (n = 1) contains zero triangles, indicating that the correct expression must account for one less than k.

D ((k + 2)^2) YOUR ANSWER

This expression indicates an even higher count of triangles than (k + 1)^2, suggesting that there are two additional triangles compared to the expected count. Such an increase does not correspond to the established pattern, which reflects a decrease of one triangle relative to k.

Conclusion

The expression (k - 1)^2 correctly represents the number of equilateral triangles in a figure when n = k, reflecting a decrease in triangle count as n increases from the base case of n = 1. The incorrect options fail to account for this decrease or suggest an inaccurate relationship between n and the triangle count, demonstrating the importance of understanding the specific growth pattern in the series.

Each of the figures shown, except for n = 1, is composed of one or more small, identical equilateral triangles. If this pattern continues, which of the following expressions would represent the number of these equilateral triangles in a figure when n = k?

(k - 1)^2

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