Chapter 1.6, Problem 36IP

Solution

(a)

To write the set of ordered pairs.

  $\left\{\left(1,4\right),\left(2,7\right),\left(3,10\right),\left(4,13\right)\right\}$

GIVEN:

Term number	1	2	3	4
Term	4	7	10	13

Explanation:

To write the set of ordered pairs, the term number given is x-coordinate and the term is y-coordinate.

The set of ordered pairs is:

  $\left\{\left(1,4\right),\left(2,7\right),\left(3,10\right),\left(4,13\right)\right\}$

(b)

To plot the graph of ordered pairs.

  

GIVEN:

Term number	1	2	3	4
Term	4	7	10	13

Concept Used:

In rectangular coordinate system any poin be in the form of $\left(x,y\right)$ .

• In Quadrant I, both the coordinates of ordered pair are positive, that is, $\left(+,+\right)$ .
• In Quadrant II, first number is negative and second is positive, that is, $\left(-,+\right)$ .
• In Quadrant III, both the coordinates of ordered pair are negative, that is, $\left(-,-\right)$ .
• In Quadrant IV, first point is positive and second is negative, that is, $\left(+,-\right)$ .

Explanation: In order to plot the point $\left(1,2\right)$ , observe that both numbers of the ordered pair are positive, so the point will be in first quadrant.To plot the graph of the point, first move 1 units to the right starting from origin and then move 4 units up.

In order to plot the point $\left(2,7\right)$ , observe that both numbers of the ordered pair are positive, so the point will be in first quadrant.To plot the graph of the point, first move 2 units to the right starting from origin and then move 7 units up.

In order to plot the point $\left(3,10\right)$ , observe that both numbers of the ordered pair arepositive, so the point will be in first quadrant.To plot the graph of the point, first move 3 units to the right starting from origin and then move 10 units up.

In order to plot the point $\left(4,13\right)$ , observe that both numbers of the ordered pair arepositive, so the point will be in first quadrant.To plot the graph of the point, first move 4 units to the right starting from origin and then move 13 units up

The graph of the points is shown below:

  

(c)

To describe the shape of graph.

The graph shows a positive, linear relationship.

(d)

To find the 20th term of arithmetic sequence, and the rule to find the 20^th term.

  ${\text{a}}_{\text{n}}={\text{a}}_1+\left(\text{n}-1\right)\text{d}$

20^th term is 61

GIVEN:

Term number	1	2	3	4
Term	4	7	10	13

In order to find the rule and the 20^th term, it is an arithmetic sequence, and nth term of arithmetic sequence is given by the formula:

  ${\text{a}}_{\text{n}}={\text{a}}_1+\left(\text{n}-1\right)\text{d}$

where ${\text{a}}_{\text{n}}$ is nth term, ${\text{a}}_1$ is first term, and dis common difference.

Now in order to find 20^th term by using the formula ${\text{a}}_{\text{n}}={\text{a}}_1+\left(\text{n}-1\right)\text{d}$

To simplify further as shown below:

  $\begin{array}{l}{\text{a}}_{\text{n}}={\text{a}}_1+\left(\text{n}-1\right)\text{d}\\ {\text{a}}_1=4\\ \text{d}=3\\ \text{n}=20\\ {\text{a}}_{20}=4+\left(20-1\right)3\\ {\text{a}}_{20}=61\end{array}$

So, the 20^th term is 61