Final answer:
1. Example: n = 0, 0² + 1 = 1
2. Example: n = 1, 1² + 1 = 3
3. Example: n = -2, (-2)² + 1 = 5
4. Example: n = 5, 5² + 1 = 26
Step-by-step explanation:
The statement "For every integer n, the value of n² + 1" is a conditional statement that asserts that for any integer value of n, the expression n² + 1 will have a particular value. In this explanation, we will provide four examples that satisfy this condition and also a counterexample that does not satisfy it.
Example 1: When n is zero, the expression simplifies to 0² + 1, which is equal to one. Therefore, this example satisfies the condition for all integer values of n when n is zero.
Example 2: When n is one, the expression simplifies to 1² + 1, which is equal to three. Therefore, this example satisfies the condition for all integer values of n when n is one.
Example 3: When n is negative two, the expression simplifies to (-2)² + 1, which is equal to five. Therefore, this example satisfies the condition for all integer values of n when n is negative two.
Example 4: When n is five, the expression simplifies to 5² + 1, which is equal to twenty-six. Therefore, this example satisfies the condition for all integer values of n when n is five.
However, there exists a counterexample where the condition does not hold true. For instance, let us consider the value of n as -3. When we substitute this value in the expression, we get (-3)² + 1 which simplifies to nine. This value does not satisfy the given condition as it is not equal to any integer value for any integer value of n when n is -3. Therefore, we can say that while the given statement holds true for most integer values of n, it does not hold true for all integer values of n.