Question

use 100 tiles
stage 1\tstage 2\tstage\tborder toothpicks\tborder tiles
\t\t1\t8\t12
\t\t2\t16\t20
stage 3\t\t3\t24\t28
?\ta. one stage uses 100 border tiles. which stage?

Explanation:

Step1: Identify the pattern for Border Tiles

Looking at the table:

• Stage 1: 12 border tiles
• Stage 2: 20 border tiles
• Stage 3: 28 border tiles

The difference between consecutive stages: \(20 - 12 = 8\), \(28 - 20 = 8\). So it's an arithmetic sequence with first term \(a_1 = 12\) and common difference \(d = 8\).

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n - 1)d\).

Step2: Substitute values and solve for \(n\)

We know \(a_n = 100\), \(a_1 = 12\), \(d = 8\).

Substitute into the formula:
\[
100 = 12 + (n - 1) \times 8
\]

Subtract 12 from both sides:
\[
100 - 12 = (n - 1) \times 8
\]
\[
88 = (n - 1) \times 8
\]

Divide both sides by 8:
\[
n - 1 = \frac{88}{8} = 11
\]

Add 1 to both sides:
\[
n = 11 + 1 = 12
\]

Answer:

12