Chapter 8.1, Problem 12E

(a).

To determine

To find the common difference.

Answer to Problem 12E

The common difference is $-2$ .

Explanation of Solution

Given:

Sequence, $15,13,11,9,\dots$

Concept Used:

A sequence $\left\{a_n\right\}$ is an arithmetic sequence if it can be written in the form $\left\{a,a+d,a+2d,\dots ,a+\left(n-1\right)d,\dots \right\}$ for some constant $d$ ,the number $d$is the common difference.

Calculation:

By comparing the arithmetic sequence:

  $\begin{array}{c}a+d=13,\\ a=15\end{array}$

The common difference is:

  $\begin{array}{c}\left(a+d\right)-a=13-15\\ =-2\end{array}$

Conclusion:

The common difference is $-2$ .

(b).

To determine

To find: The $\text{8th}$ term.

Answer to Problem 12E

The $\text{8th}$ term is $106$ .

Explanation of Solution

Given:

Sequence, $15,13,11,9,\dots$

Concept Used:

• The $n^{th}$ is:

$a_n=a+\left(n-1\right)d$

Where, $a$ =First term, $d$ =common difference and $n$ = $n^{th}$ term.

Calculation:

Here,

  $a=15$ ,

  $d=-2$

  $\text{8th}$ term is:

  $\begin{array}{c}{a}_8=15+\left(8-1\right)13\\ =15+7\left(13\right)\\ =106\end{array}$

Conclusion:

The $\text{8th}$ term is $106$

(c).

To determine

To find: A recursive rule for $n^{th}$ term.

Answer to Problem 12E

A recursive rule for $n^{th}$ term.is $a_n=a_{n-1}-2$

Explanation of Solution

Given:

Sequence, $15,13,11,9,\dots$

Concept Used:

Each term is an arithmetic sequence can be obtained recursively from its preceding term by adding $d$ :

  $a_n=a_{n-1}+d$ for all $n\ge 2$ .

Calculation:

Here,

  $a=15$ ,

  $d=-2$

By using recursive rule, $n^{th}$ is:

  $a_n=a_{n-1}-2$

Conclusion:

A recursive rule for $n^{th}$ term.is $a_n=a_{n-1}-2$

(d).

To determine

To find an explicit rule for $n^{th}$ term.

Answer to Problem 12E

An explicit rule for $n^{th}$ term is $a_n=13n+2$ .

Explanation of Solution

Given:

Sequence, $15,13,11,9,\dots$

Concept Used:

The rule for explicit $n^{th}$ is:

  $a_n=a+\left(n-1\right)d$

Where, $a$ =First term, $d$ =common difference and $n$ = $n^{th}$ term.

Calculation:

Here,

  $a=15$

The explicit rule is:

  $\begin{array}{c}{a}_n=15+\left(n-1\right)\left(13\right)\\ =15+13n-13\\ =13n+2\end{array}$

Conclusion:

An explicit rule for $n^{th}$ term is $a_n=13n+2$ .