Chapter 12, Problem 14RE

a.

To determine

To find: the first five terms of the sequence.

Answer to Problem 14RE

The first five terms of the sequence are:

  $3.5,3,2.5,2,1.5$

Explanation of Solution

Given information: Given sequence is $a_n=4-\frac{n}{2}$ .

Calculation:

The first five terms of the sequence are:

  $\begin{array}{c}{a}_n=4-\frac{n}{2}.\end{array}$

  $\begin{array}{c}n=1\to a_1=4-\frac{1}{2}=\frac{8-1}{2}=\frac{7}{2}=\mathrm{3.5.}\end{array}$

  $\begin{array}{c}n=2\to a_2=4-\frac{2}{2}=4-1=3.\end{array}$

  $\begin{array}{c}n=3\to a_3=4-\frac{3}{2}=\frac{8-3}{2}=\frac{5}{2}=\mathrm{2.5.}\end{array}$

  $\begin{array}{c}n=4\to a_4=4-\frac{4}{2}=4-2=2.\end{array}$

  $\begin{array}{c}n=5\to a_1=4-\frac{5}{2}=\frac{8-5}{2}=\frac{3}{2}=\mathrm{1.5.}\end{array}$

So, the first five terms of the sequence are:

  $3.5,3,2.5,2,1.5$

b.

To determine

To graph: the terms found in part (a).

Answer to Problem 14RE

Explanation of Solution

Given information: Given sequence is $a_n=4-\frac{n}{2}$ .

Calculation:

The table on the given sequence is shown below.

n	$a_n$
1	3.5
2	3
3	2.5
4	2
5	1.5

The graph on the above table of first five terms of the given sequence is shown below.

  

c.

To determine

To find: the fifth partial sum of the sequence.

Answer to Problem 14RE

The fifth partial sum of the sequence=12.5.

Explanation of Solution

Given information: Given sequence is $a_n=4-\frac{n}{2}$ .

Calculation:

The first five terms of the sequence are:

  $3.5,3,2.5,2,1.5$

The fifth partial sum of the sequence=sum of first five term of the sequence.

The fifth partial sum of the sequence=3.5+3+2.5+2+1.5=12.5.

So, the fifth partial sum of the sequence=12.5.

d.

To determine

To find: whether the series is arithmetic or geometric and the common difference or the common ratio.

Answer to Problem 14RE

The given sequence is arithmetic with common difference=-0.5.

Explanation of Solution

Given information: Given sequence is $a_n=4-\frac{n}{2}$ .

Calculation:

The first five terms of the sequence are:

  $3.5,3,2.5,2,1.5$

Therefore,

  $\begin{array}{l}{a}_2-a_1=3-3.5=-\mathrm{0.5.}\\ a_3-a_2=2.5-3=-\mathrm{0.5.}\\ a_4-a_3=2-2.5=-\mathrm{0.5.}\\ a_5-a_4=1.5-2=-\mathrm{0.5.}\end{array}$

Therefore, the given sequence is arithmetic with common difference=-0.5.