Question

the first figure of the sierpinski triangle has one shaded triangle. the second figure of the sierpinski triangle has three shaded triangles. the third figure of the sierpinski triangle has nine shaded triangles. which summation represents the total number of shaded triangles in the first 15 figures?
sum_{n = 1}^{15}1(3)^{n - 1}
sum_{n = 1}^{15}3(1)^{n - 1}
sum_{n = 1}^{15}1left(\frac{1}{3}
ight)^{n - 1}
sum_{n = 1}^{15}\frac{1}{3}(1)^{n - 1}

Explanation:

Step1: Identify the geometric - sequence pattern

The number of shaded triangles in the Sierpinski - triangle figures forms a geometric sequence. The first term \(a_1 = 1\), the second term \(a_2=3\), and the third term \(a_3 = 9\). The common ratio \(r=\frac{a_{n + 1}}{a_n}\), so \(r=\frac{3}{1}=3\).

Step2: Recall the sum formula for a geometric series

The sum of the first \(n\) terms of a geometric series is given by \(S_n=\sum_{k = 1}^{n}a_1r^{k - 1}\), where \(a_1\) is the first - term and \(r\) is the common ratio. Here, \(n = 15\), \(a_1 = 1\), and \(r = 3\).

Answer:

\(\sum_{n = 1}^{15}1(3)^{n - 1}\)