The problem involves an arithmetic sequence, where the first two terms are $p + 6$ and $24$, and the fourth term is $18$. We are to find the value of $p$.

Step-by-step solution:

1. Arithmetic sequence definition:
   The difference $d$ between consecutive terms in an arithmetic sequence is constant. Let the terms of the sequence be: $T_1 = p + 6, \quad T_2 = 24, \quad T_3 = T_2 + d, \quad T_4 = T_3 + d.$

2. Difference between the first two terms:
   The common difference $d$ can be found using the first two terms: $d = T_2 - T_1 = 24 - (p + 6) = 24 - p - 6 = 18 - p.$

3. Expression for the fourth term:
   The fourth term is: $T_4 = T_3 + d = (T_2 + d) + d = T_2 + 2d.$ Substituting $T_2 = 24$: $T_4 = 24 + 2d.$

4. Substitute $T_4 = 18$:
   Given that the fourth term $T_4$ is $18$: $18 = 24 + 2d.$ Simplify: $18 - 24 = 2d \implies -6 = 2d \implies d = -3.$

5. Substitute $d = -3$ to find $p$:
   Using $d = 18 - p$: $-3 = 18 - p.$ Solve for $p$: $p = 18 + 3 = 21.$

Final Answer:

$p = 21.$

Would you like a detailed explanation of any step?

Related Questions:

1. How do we derive the formula for the $n$-th term in an arithmetic sequence?
2. If $p = 21$, what are the first five terms of the sequence?
3. What happens to the sequence if $d$ is positive instead of negative?
4. How do we calculate the sum of the first $n$ terms in an arithmetic sequence?
5. Can arithmetic sequences have fractional or irrational common differences?

Tip:

Always verify your solution by substituting back into the original sequence to ensure consistency!