Answered

Get personalized answers to your specific questions with IDNLearn.com. Whether it's a simple query or a complex problem, our experts have the answers you need.

Instructions: Show that the conjecture is false by finding a counterexample. Change the value of [tex] n [/tex] below in order to make it a counterexample.

"For every integer [tex] n [/tex], [tex] n^3 [/tex] is positive."

[tex]\[
\begin{array}{l}
n=1 \\
n=-3 \\
n=2 \\
n=5
\end{array}
\][/tex]

[tex]\[
n= \, \_\_\_\_
\][/tex]

Sagot :

GinnyAnsweridn
To address the conjecture "For every integer [tex]\( n \)[/tex], [tex]\( n^3 \)[/tex] is positive," we will explore the values of [tex]\( n \)[/tex] given in the question:

1. [tex]\( n = 1 \)[/tex]
2. [tex]\( n = -3 \)[/tex]
3. [tex]\( n = 2 \)[/tex]
4. [tex]\( n = 5 \)[/tex]

We will calculate [tex]\( n^3 \)[/tex] for each value of [tex]\( n \)[/tex]:

1. For [tex]\( n = 1 \)[/tex]:
[tex]\[ 1^3 = 1 \][/tex]
The result is positive.

2. For [tex]\( n = -3 \)[/tex]:
[tex]\[ (-3)^3 = -27 \][/tex]
The result is negative.

3. For [tex]\( n = 2 \)[/tex]:
[tex]\[ 2^3 = 8 \][/tex]
The result is positive.

4. For [tex]\( n = 5 \)[/tex]:
[tex]\[ 5^3 = 125 \][/tex]
The result is positive.

From these calculations, we can see that [tex]\( n = -3 \)[/tex] results in [tex]\( (-3)^3 = -27 \)[/tex], which is a negative number. This serves as a counterexample that disproves the conjecture.

Therefore, we have demonstrated that the conjecture "For every integer [tex]\( n \)[/tex], [tex]\( n^3 \)[/tex] is positive" is false. The specific value of [tex]\( n \)[/tex] that serves as the counterexample is [tex]\( n = -3 \)[/tex].

So, the value of [tex]\( n \)[/tex] that makes the conjecture false is:
[tex]\[ n = -3 \][/tex]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your questions deserve accurate answers. Thank you for visiting IDNLearn.com, and see you again for more solutions.