Consider the following linear number patterns: 23; 17; 11; p; -1; ext and 2x+1, 18, 4x+5. Find the value of p and x.

For a linear number pattern, the common difference between consecutive terms is constant.

$$d = T_2 - T_1 = 17 - 23 = -6$$

The common difference is $-6$.

The term $p$ is the term after 11.

$$p = 11 + d$$ $$p = 11 + (-6)$$ $$p = \boxed{5}$$

3.1.2: Determine the $n^{th}$ term of the number pattern

The formula is $T_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.
Here, $a = 23$ and $d = -6$.

$$T_n = 23 + (n-1)(-6)$$ $$T_n = 23 - 6n + 6$$ $$T_n = \boxed{-6n + 29}$$

3.1.3: Determine the $15^{th}$ term of the sequence

$$T_{15} = -6(15) + 29$$ $$T_{15} = -90 + 29$$ $$T_{15} = \boxed{-61}$$

3.1.4: Which term will be equal to -121?

$$-6n + 29 = -121$$ $$-6n = -121 - 29$$ $$-6n = -150$$ $$n = \frac{-150}{-6}$$ $$n = \boxed{25}$$

The $25^{th}$ term will be equal to $-121$.

Part 3.2: $2x+1, 18, 4x+5$ are the first 3 terms of a linear number pattern

3.2.1: Determine the value of $x$

In a linear pattern, the common difference between consecutive terms is constant.
So, $T_2 - T_1 = T_3 - T_2$.

$$18 - (2x+1) = (4x+5) - 18$$

$$18 - 2x - 1 = 4x + 5 - 18$$ $$17 - 2x = 4x - 13$$ $$17 + 13 = 4x + 2x$$ $$30 = 6x$$ $$x = \frac{30}{6}$$ $$x = \boxed{5}$$