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]
