Chapter 5.4, Problem 39PPS

(a)

To determine

To Find: The expression for the sum of the cube of the number line.

Answer to Problem 39PPS

The expression for the sum of the cube of the number and the three is $n^3+3$ .

Explanation of Solution

Consider that the number $n$ is represented as $n^3$ .

Thus, the expression for the sum of the cube of the number and the three is $n^3+3$ .

(b)

To determine

To Find: The table that includes the value of the expression for the 10 different values of $n$ that include both the even values of n .

Answer to Problem 39PPS

The required table is shown in Table 1

Explanation of Solution

Consider the given expression is $n^3+3$ .

The table for the value is shown in Table 1

Table 1

$n$	$n^3+3$
-4	-61
-3	-24
-2	-5
-1	2
0	3
1	4
2	11
3	30
4	67
5	128

(c)

To determine

To Find: The conjecture about the values of the n when the value of the expression is even.

Answer to Problem 39PPS

The conjecture is if $n^3+3$ even then the n is odd.

Explanation of Solution

Consider the given expression is $n^3+3$ .

From the table 1 it is clear that the expression $n^3+3$ is even when the value of $n$ is odd.

Thus, the conjecture is if $n^3+3$ even then the n is odd.

(d)

To determine

To Find: The indirect proof of the conjecture.

Answer to Problem 39PPS

The assumption is false.

Explanation of Solution

Calculation:

Consider that for the direct proof it must be assumed temporarily that the conclusion is false. It is observed by assuming that it is logically impossible to prove that the assumption is false as the original conclusion is true. For part (d) as the value fo $n$ is even.

Consider the expression as $n^3+3$ is even then $n$ is odd is to be proved.

Consider the indirect proof that he n is even then,

  $\begin{array}{c}{\left(2k\right)}^3+3=8k^3+3\\ =2\left(4k^3+1\right)+1\end{array}$

Consider that for kto be the integer $4k^3+1$ is an integer. Thus, the $n^3+3=2\left(4k^2+1\right)+1$ is odd that contradicts the hypotheses that $n^3+3$ is even.

Thus, the assumption is false.