10) make a table of values for the rule x2 + x + 11 when x is an integer from 1 to 8. make a conjecture about the type of number generated by the rule. continue your table. what value of x generates a counterexample?

We have been given a rule and we are required to prepare a table of values using this rule and values of x from 1 to 8.

For x=1, we get:

For x=2, we get:

For x=3, we get:

For x=4, we get:

For x=5, we get:

For x=6, we get:

For x=7, we get:

For x=8, we get:

We can see that all these values are prime numbers. So, we can make a conjecture that the given rule produces prime numbers.

Now we need to find a counter example, that is, a value of x that produces a non-prime number, that is, a composite number.

Let us calculate the value of the given function at x=10. We get the value of the function as:

For x=10, we get:

We know that 121 is a composite number as it is divisible by 11.

Therefore, x=10 generates a counter example.

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