Chapter 8, Problem 1Q

To determine

Find the pattern, next term and write a rule for the $n\text{th}$ term of the given sequence.

Answer to Problem 1Q

The pattern is arithmetic.

The rule for the $n\text{th}$ layer is $a_n=1+\left(n-1\right)6$ .

The $5\text{th}$ term is $a_5=25$ .

Explanation of Solution

Given:

The given sequence is $1,7,13,19,---$ .

Calculation:

The difference of consecutive terms is

  $\begin{array}{l}7-1=6\\ 13-7=6\\ 19-13=6\end{array}$

In an arithmetic sequence, the difference of consecutive terms, called the common difference, is constant.

So, the sequence is arithmetic.

The general rule for the $n\text{th}$ term is $a_n=a_1+\left(n-1\right)d$ .

Where $a_1\to$ first term, $d\to$ common difference, $n\to$ number of terms.

  $a_1\to 1$

  $d\to 6$

The $n\text{th}$ term is $a_n=1+\left(n-1\right)6$

The $5\text{th}$ term is

  $\begin{array}{l}{a}_5=1+\left(5-1\right)6\\ \Rightarrow a_5=1+24\\ \Rightarrow a_5=25\end{array}$

Hence

The pattern is arithmetic.

The rule for the $n\text{th}$ layer is

  $a_n=1+\left(n-1\right)6$ .

The $5\text{th}$ term is $a_5=25$ .