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Question

# Make a table of values for the rule x$^2+x+11$ 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?

Solution

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Given expression:

$\begin{equation*} x^2 + x + 11 \end{equation*}$

Substitute $x=1$ and evaluate the value of the expression:

$\begin{equation*} (1)^2 + 1 + 11=13 \end{equation*}$

Substitute $x=2$ and evaluate the value of the expression:

$\begin{equation*} (2)^2 + 2 + 11=17 \end{equation*}$

Substitute $x=3$ and evaluate the value of the expression:

$\begin{equation*} (3)^2 + 3 + 11=23 \end{equation*}$

Substitute $x=4$ and evaluate the value of the expression:

$\begin{equation*} (4)^2 + 4 + 11=31 \end{equation*}$

Substitute $x=5$ and evaluate the value of the expression:

$\begin{equation*} (5)^2 + 5 + 11=41 \end{equation*}$

Substitute $x=6$ and evaluate the value of the expression:

$\begin{equation*} (6)^2 + 6 + 11=53 \end{equation*}$

Substitute $x=7$ and evaluate the value of the expression:

$\begin{equation*} (7)^2 + 7 + 11=67 \end{equation*}$

Substitute $x=8$ and evaluate the value of the expression:

$\begin{equation*} (8)^2 + 8 + 11=83 \end{equation*}$

This expression generates prime numbers in ascending order.

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