Chapter 8.1, Problem 13E

(a).

To determine

To find the common difference.

Answer to Problem 13E

The common difference is $\frac{1}{2}$

Explanation of Solution

Given:

Sequence, $1,\frac{3}{2},2,\frac{5}{2},\dots$

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:

  $\begin{array}{c}a+d=\frac{3}{2},\\ a=1\end{array}$

The common difference is:

  $\begin{array}{c}\left(a+d\right)-a=\frac{3}{2}-1\\ =\frac{1}{2}\end{array}$

Conclusion:

The common difference is $\frac{1}{2}$ .

(b).

To determine

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

Answer to Problem 13E

The eight term is $\frac{9}{2}$

Explanation of Solution

Given:

  $1,\frac{3}{2},2,\frac{5}{2},\dots$

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,

  $a=1$ ,

  $d=\frac{1}{2}$

  $\text{8th}$ term is calculated as follow:

  $\begin{array}{c}{a}_8=1+\left(8-1\right)\left(\frac{1}{2}\right)\\ =1+7\left(\frac{1}{2}\right)\\ =\frac{9}{2}\end{array}$

Conclusion:

The $\text{8th}$ term is $\frac{9}{2}$

(c).

To determine

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

Answer to Problem 13E

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

Explanation of Solution

Given:

Sequence, $1,\frac{3}{2},2,\frac{5}{2},\dots$

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,

  $a=1$ ,

  $d=\frac{1}{2}$

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

  $a_n=a_{n-1}+\frac{1}{2}$

Conclusion:

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

(d).

To determine

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

Answer to Problem 13E

An explicit rule for $n^{th}$ term is $a_n=\frac{\left(n+1\right)}{2}$ .

Explanation of Solution

Given:

Sequence, $1,\frac{3}{2},2,\frac{5}{2},\dots$

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,

  $a=1$ ,

  $d=\frac{1}{2}$

The explicit rule is:

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

Conclusion:

An explicit rule for $n^{th}$ term is $a_n=\frac{\left(n+1\right)}{2}$ .