Let's break down the problem based on the information provided:

Given Sequence:

• The sequence starts with: $-21, -18, -15, -12, \ldots$

2.1. General Term of the Sequence

The sequence is an arithmetic progression (AP) where the first term $a_1 = -21$ and the common difference $d = -18 - (-21) = 3$.

The general term $a_n$ of an arithmetic sequence is given by the formula: $a_n = a_1 + (n - 1) \times d$ Substituting the values: $a_n = -21 + (n - 1) \times 3$ Simplifying: $a_n = 3n - 24$

2.2. Value of the 100th Term

To find the 100th term $a_{100}$, substitute $n = 100$ into the general term formula: $a_{100} = 3(100) - 24 = 300 - 24 = 276$

2.3. Checking if 0 is a Term of the Sequence

We need to check if there exists an $n$ such that $a_n = 0$. Set the general term equal to 0: $3n - 24 = 0$ Solving for $n$: $3n = 24 \quad \Rightarrow \quad n = \frac{24}{3} = 8$ Since $n = 8$ is a positive integer, 0 is indeed a term of the sequence (the 8th term).

2.4. Sum of the First Fifteen Terms

The sum $S_n$ of the first $n$ terms of an arithmetic sequence is given by: $S_n = \frac{n}{2} \times (a_1 + a_n)$ Here, $n = 15$, $a_1 = -21$, and $a_{15}$ can be found using the general term formula: $a_{15} = 3(15) - 24 = 45 - 24 = 21$ Now, calculate the sum $S_{15}$: $S_{15} = \frac{15}{2} \times (-21 + 21) = \frac{15}{2} \times 0 = 0$

Thus, the sum of the first fifteen terms is $0$.

Summary of Answers:

1. General Term: $a_n = 3n - 24$
2. 100th Term: $a_{100} = 276$
3. 0 as a Term: Yes, $0$ is the 8th term.
4. Sum of the First Fifteen Terms: $0$

Would you like more details on any specific part of the solution?

Here are some questions that might expand your understanding:

1. How would you derive the general term for other sequences?
2. Why is the common difference important in an arithmetic sequence?
3. What happens if the common difference is negative?
4. How do you verify if a specific number is part of a sequence?
5. How is the sum of the first $n$ terms different for geometric sequences?
6. Can the general term formula be rearranged to solve for $n$?
7. What if the sequence had a fractional common difference?
8. How would the sum change if more terms were added?

Tip: Always verify if the value of $n$ is a positive integer when checking if a specific number is part of a sequence.