Chapter 8.1, Problem 14E

(a).

To determine

To find the common difference.

Answer to Problem 14E

The common difference is $0.1$

Explanation of Solution

Given:

Sequence, $3,3.1,3.2,3.3,\dots$

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:

  $\begin{array}{c}a+d=3.1,\\ a=3\end{array}$

The common difference is:

  $\begin{array}{c}\left(a+d\right)-a=3.1-3\\ =0.1\end{array}$

Conclusion:

The common difference is $0.1$

(b).

To determine

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

Answer to Problem 14E

The $\text{8th}$ term is $7.7$

Explanation of Solution

Given:

Sequence, $3,3.1,3.2,3.3,\dots$

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,

  $a=3$ ,

  $d=0.1$

  $\text{8th}$ term is:

  $\begin{array}{c}{a}_8=3+\left(8-1\right)\left(0.1\right)\\ =7+7\left(0.1\right)\\ =7.7\end{array}$

Conclusion:

The $\text{8th}$ term is $7.7$

(c).

To determine

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

Answer to Problem 14E

A recursive rule for $n^{th}$ term.is $a_n=a_{n-1}+0.1$

Explanation of Solution

Given:

Sequence, $3,3.1,3.2,3.3,\dots$

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,

  $a=3$ ,

  $d=0.1$

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

  $a_n=a_{n-1}+0.1$

Conclusion:

A recursive rule for $n^{th}$ term.is $a_n=a_{n-1}+0.1$

(d).

To determine

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

Answer to Problem 14E

An explicit rule for $n^{th}$ term is $a_n=\frac{n+29}{10}$ .

Explanation of Solution

Given:

Sequence, $3,3.1,3.2,3.3,\dots$

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,

  $a=3$ ,

  $d=0.1$

The explicit rule is:

  $\begin{array}{c}{a}_n=3+\left(n-1\right)\left(0.1\right)\\ =3+0.1n-0.1\\ =0.1n+2.9\\ =\frac{n+29}{10}\end{array}$

Conclusion:

An explicit rule for $n^{th}$ term is $a_n=\frac{n+29}{10}$ .