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 small, identical equilateral triangles in the figure when n = k, based on the observed pattern of triangle arrangement as n increases.

A ((k - 1)^2) CORRECT

This expression correctly captures the number of equilateral triangles for the given pattern, as it indicates that for each increment in n, the number of triangles increases in a manner consistent with the square of one less than the current n value.

B (k^2) YOUR ANSWER

The expression k^2 implies that the number of triangles grows quadratically with n, starting from n = 1. However, this does not align with the observed pattern, which shows that at n = 1, there are no triangles, and the count only begins to increase from n = 2 onward.

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

This choice suggests that the triangle count would increase by a square of one more than n, which is inconsistent with the given pattern. The count does not start from a value that increases quadratically but rather from a base case that leads to a different growth pattern.

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

Using (k + 2)^2 implies an even larger triangle count that does not match the observed sequence. This expression suggests a base of 4 triangles at n = 0, which contradicts the foundational premise of the problem where no triangles exist at n = 1.

Conclusion

The expression (k - 1)^2 is the only option that accurately represents the number of equilateral triangles in the pattern for n = k, starting from zero triangles at n = 1 and aligning with the incremental increases observed in subsequent figures. Other options either misrepresent the base case or suggest incorrect growth patterns, failing to capture the essential relationship between n and the number of triangles.

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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