Question

assume the following pattern with terms built of square tile figures continues and complete parts (a) through (c) below.
a. how many square tiles are there in the eighth figure?
b. how many square tiles are in the nth figure?
a. the figures do not appear to follow any predictable pattern.
b. there are 3n - 1 square tiles in the nth figure.
c. there are 3n square tiles in the nth figure.
d. there are 3n + 1 square tiles in the nth figure.
c. is there a figure that has exactly 524 square tiles? if so, which one? select the correct choice below and, if necessary, fill in the answer box to
a. yes. figure number has 524 square tiles.
b. no. there does not exist a figure with exactly 524 square tiles.

Explanation:

Part (a)

Step1: Identify the pattern

Count the number of tiles in each figure:

• Figure 1: 2 tiles
• Figure 2: 5 tiles
• Figure 3: 8 tiles
• Figure 4: 11 tiles

The pattern is an arithmetic sequence with first term \(a_1 = 2\) and common difference \(d = 3\). The formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\). Substituting \(a_1 = 2\) and \(d = 3\), we get \(a_n=2+(n - 1)\times3=3n - 1\).

Step2: Calculate for \(n = 8\)

Substitute \(n = 8\) into \(a_n=3n - 1\):
\(a_8=3\times8 - 1=24 - 1 = 23\)

From the pattern analysis in part (a), we derived the formula for the \(n\)-th term as \(a_n = 3n-1\). Let's verify with the given figures:

• For \(n = 1\): \(3\times1-1 = 2\) (matches Figure 1)
• For \(n = 2\): \(3\times2-1 = 5\) (matches Figure 2)
• For \(n = 3\): \(3\times3-1 = 8\) (matches Figure 3)
• For \(n = 4\): \(3\times4-1 = 11\) (matches Figure 4)

So the correct option is B.

We use the formula \(a_n=3n - 1\) and set it equal to 524:
\(3n-1 = 524\)

Step1: Solve for \(n\)

Add 1 to both sides: \(3n=524 + 1=525\)

Divide both sides by 3: \(n=\frac{525}{3}=175\)

Since \(n = 175\) is a positive integer, there exists a figure (Figure 175) with 524 tiles.

Answer:

23

Part (b)