Chapter 1.3, Problem 44HP

To determine

Write two different rules that relate the figure number to the number of toothpicks in each figure.

Answer to Problem 44HP

Figure 1: 6 toothpicks; Figure 2: 8 toothpicks; Figure 3: 10 toothpicks

Explanation of Solution

Given:

Write two different rules that relate the figure number to the number of toothpicks in each figure. Explain how you arrived at your answer.

Concept Used:

Figure 1: 6 toothpicks

Figure 2: 8 toothpicks

Figure 3: 10 toothpicks

The common difference between two consecutive figure is 8 − 6 = 2 toothpicks.

This is an arithmetic sequence.

Calculation:

Explicit formula for Arithmetic sequence: $a_n=a_1+d(n-1)$

The first term $a_1$ is the first figure: 6 toothpicks, we can write $a_1$ = 6 and the common difference d = 2

Let we find the number of toothpicks in the third figure by using the explicit formula

  $\begin{array}{l}{a}_n=a_1+d(n-1)\\ a_3=a_1+d(3-1)=6+2(3-1)=6+4=10\end{array}$

So, the number of toothpicks in the 3^rd figure is 8.

2^nd way we can find the number of toothpicks is the Recursive Formula for Arithmetic Sequence.

Recursive formula: $a_n=a_{n-1}+d$

The first term $a_1$ is the first figure: 6 toothpicks, we can write $a_1$ = 6 and the common difference d = 2

Let we find the number of toothpicks in the third figure by using the explicit formula $a_n$ is the number of toothpicks of 3^rd figure and $a_{\mathrm{n-1}}$ is the number of toothpicks of 2^nd figure

We know that $a_{\mathrm{n-1}}$= 8 and d = 2; $a_n=a_{n-1}+d\Rightarrow a_3=a_2+d\Rightarrow a_3=6+2=10$

So, the number of toothpicks in the 3^rd figure is 8.

Thus, Figure 1: 6 toothpicks; Figure 2: 8 toothpicks; Figure 3: 10 toothpicks