Question

which explicit formula can be used to determine the number of triangles in figure n?
figure 1 figure 2 figure 3
( a _ { n } = 2 ( 4 ) ^ { n - 1 } )
( a _ { n } = 4 + 2 n )
( a _ { n } = 4 n )
( a _ { n } = 4 ( 2 ) ^ { n - 1 } )

Explanation:

Step1: Analyze the number of triangles in each figure

• Figure 1: \(a_1 = 4\)
• Figure 2: \(a_2 = 8\)
• Figure 3: \(a_3 = 16\)

Step2: Check the general formula for a geometric sequence

The general formula for a geometric sequence is \(a_n=a_1r^{n - 1}\), where \(a_1\) is the first - term and \(r\) is the common ratio.
Here, \(a_1 = 4\) and \(r=\frac{a_2}{a_1}=\frac{8}{4} = 2\)

Step3: Substitute \(a_1\) and \(r\) into the formula

Substituting \(a_1 = 4\) and \(r = 2\) into \(a_n=a_1r^{n - 1}\), we get \(a_n=4\times(2)^{n - 1}\)

Answer:

\(a_n = 4(2)^{n-1}\) (the fourth option)