31. 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?
Answer: A
The expression that represents the number of equilateral triangles in a figure when n = k is (k - 1)^2.
The expression (k - 1)^2 accurately represents the number of small, identical equilateral triangles in the figure for any value of n greater than 1.
A) (k - 1)^2
This option is correct as it represents the number of equilateral triangles when n = k. The pattern indicates that for each increase in n, the number of triangles is determined by squaring (k - 1), which aligns with the given sequence and allows for the correct calculation of triangles based on the defined pattern.
B) k^2
This option is incorrect because it suggests that the number of triangles directly corresponds to the square of k, without accounting for the initial condition that n starts from 2, which would not fit the established pattern observed in the figures.
C) (k + 1)^2
This option is also incorrect as it implies that the number of triangles increases by an additional layer, which does not correspond to the observed pattern. Instead, it suggests a total that exceeds the actual count of triangles represented for the figure when n = k.
D) (k + 2)^2
Similar to option C, this choice is incorrect because it incorrectly adds two additional layers of triangles, which deviates from the established formula. The pattern indicates a decrease in the count relative to k rather than an increase, making this option invalid.
Conclusion
The correct expression, (k - 1)^2, accurately describes the number of equilateral triangles based on the established pattern for n values greater than 1. All other options fail to reflect the correct relationship between n and the number of triangles, either by overestimating or misrepresenting the calculations needed for accurate triangle counting.