Which formula would you use to find the total number of shaded triangles in the first 10 Sierpinski triangles?
[tex]S_n=\left(1-3^{10}\right) /(-2)[/tex]
[tex]S_n=\left(1-3^{10}\right) /(2)[/tex]
[tex]S_n=\left(1-3^{10}\right) /(3)[/tex]
Answer (2)
The problem requires finding the correct formula for the total number of shaded triangles in the first 10 Sierpinski triangles.
Recognize that the number of shaded triangles at each step forms a geometric progression with a common ratio of 3.
Apply the formula for the sum of a geometric series: S n = − 2 1 − 3 n .
Substitute n = 10 to get the formula for the first 10 Sierpinski triangles: S 10 = − 2 1 − 3 10 .
The final answer is S n = ( 1 − 3 10 ) / ( − 2 ) .
Explanation
Understanding the Problem The problem states that the total number of shaded triangles in the first 4 Sierpinski triangles is 40. We need to find a formula that gives the total number of shaded triangles in the first 10 Sierpinski triangles, given three options: S n = ( 1 − 3 10 ) / ( − 2 ) , S n = ( 1 − 3 10 ) / ( 2 ) , and S n = ( 1 − 3 10 ) / ( 3 ) .
Analyzing the Sierpinski Triangle Construction Let's analyze the Sierpinski triangle construction. At each iteration, the number of shaded triangles is multiplied by 3. The first Sierpinski triangle has 1 shaded triangle, the second has 3, the third has 9, and so on. The total number of shaded triangles in the first n Sierpinski triangles is the sum of a geometric series: S n = 1 + 3 + 3 2 + ... + 3 n − 1 .
Applying the Geometric Series Formula The sum of a geometric series is given by the formula S n = 1 − r a ( 1 − r n ) , where a is the first term and r is the common ratio. In our case, a = 1 and r = 3 . So, the sum of the first n terms is S n = 1 − 3 1 ( 1 − 3 n ) = − 2 1 − 3 n .
Finding the Formula for the First 10 Sierpinski Triangles We want to find the total number of shaded triangles in the first 10 Sierpinski triangles, so we need to find S 10 . Substituting n = 10 into the formula, we get S 10 = − 2 1 − 3 10 . This matches the first option given in the problem.
Verifying the Formula Let's verify the formula with the given information that the total number of shaded triangles in the first 4 Sierpinski triangles is 40. Using the formula, S 4 = − 2 1 − 3 4 = − 2 1 − 81 = − 2 − 80 = 40 . This confirms that our formula is correct.
Final Answer Therefore, the formula to find the total number of shaded triangles in the first 10 Sierpinski triangles is S n = − 2 1 − 3 10 .
Examples
Sierpinski triangles, and the geometric series they embody, appear in various real-world scenarios. For instance, the branching of trees or blood vessels can be modeled using fractal patterns similar to Sierpinski triangles. Understanding the geometric series helps in calculating the total length or area covered by these branching structures. This has applications in biology, where the efficiency of nutrient transport in branching networks can be analyzed, or in computer graphics, where fractal patterns are used to create realistic landscapes and textures.
The correct formula to find the total number of shaded triangles in the first 10 Sierpinski triangles is S n = − 2 1 − 3 10 , as it accurately reflects the growth pattern of shaded triangles. This stems from the properties of a geometric series where each step triples the number of shaded triangles. Hence, the chosen option is the first one provided.
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