Assume the pattern continues for the following sequence of tile figures; that is, each tile is divided into four tiles in the subsequent figure. Let S(n) be the function giving the total number of tiles in the nth figure. Find a formula for S(n) in terms of n. ΔΑ S(n) = (Type an expression using n as the variable. Simplify your answer.)
Assume the pattern continues for the following sequence of tile figures; that is, each tile is divided into four tiles in the subsequent figure. Let S(n) be the function giving the total number of tiles in the nth figure. Find a formula for S(n) in terms of n. ΔΑ S(n) = (Type an expression using n as the variable. Simplify your answer.)
Algebra and Trigonometry (6th Edition)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Analysis of Tile Patterns in Geometric Figures
Assume the pattern continues for the following sequence of tile figures; that is, each tile is divided into four tiles in the subsequent figure. Let \( S(n) \) be the function giving the total number of tiles in the n-th figure. Find a formula for \( S(n) \) in terms of \( n \).
**Sequence of Tile Figures:**
- The first figure is a single triangle.
- The second figure consists of 4 smaller triangles.
- The third figure consists of 16 even smaller triangles.
- And so on...
As depicted, this pattern demonstrates the exponential growth of the number of triangles with each step.
A visual representation of the figures:
1. **First Figure:** A single triangle.
2. **Second Figure:** This single triangle is divided into 4 smaller triangles.
3. **Third Figure:** Each of the 4 triangles from the second figure is subdivided into another 4 triangles, totaling 16 triangles.
4. **Subsequent Figures:** Continue this pattern of subdivision.
\[ \cdots \]
**Determine \( S(n) \):**
If we examine the growth pattern, where each stage involves a subdivision of each triangle into 4 new triangles, this can be identified as an exponential function.
**Formula for \( S(n) \):**
\[ S(n) = 4^{(n-1)} \]
- \( S(n) \) represents the total number of triangles in the n-th figure.
- \( n \) represents the stage number (starting from 1).
---
\[ S(n) = \_\_\_ \]
(Type an expression using \( n \) as the variable. Simplify your answer.)
Thus, the simplified expression to represent the total number of tiles in the n-th figure is \( 4^{(n-1)} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F893c97d9-09c7-4f3e-b188-bcf1f5d21bb1%2F6f6c48f0-4d22-4c92-b311-3c8d76116545%2Fstuz16d_processed.png&w=3840&q=75)
Transcribed Image Text:### Analysis of Tile Patterns in Geometric Figures
Assume the pattern continues for the following sequence of tile figures; that is, each tile is divided into four tiles in the subsequent figure. Let \( S(n) \) be the function giving the total number of tiles in the n-th figure. Find a formula for \( S(n) \) in terms of \( n \).
**Sequence of Tile Figures:**
- The first figure is a single triangle.
- The second figure consists of 4 smaller triangles.
- The third figure consists of 16 even smaller triangles.
- And so on...
As depicted, this pattern demonstrates the exponential growth of the number of triangles with each step.
A visual representation of the figures:
1. **First Figure:** A single triangle.
2. **Second Figure:** This single triangle is divided into 4 smaller triangles.
3. **Third Figure:** Each of the 4 triangles from the second figure is subdivided into another 4 triangles, totaling 16 triangles.
4. **Subsequent Figures:** Continue this pattern of subdivision.
\[ \cdots \]
**Determine \( S(n) \):**
If we examine the growth pattern, where each stage involves a subdivision of each triangle into 4 new triangles, this can be identified as an exponential function.
**Formula for \( S(n) \):**
\[ S(n) = 4^{(n-1)} \]
- \( S(n) \) represents the total number of triangles in the n-th figure.
- \( n \) represents the stage number (starting from 1).
---
\[ S(n) = \_\_\_ \]
(Type an expression using \( n \) as the variable. Simplify your answer.)
Thus, the simplified expression to represent the total number of tiles in the n-th figure is \( 4^{(n-1)} \).
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