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Given Sequence:

• The sequence provided is: $-21, -18, -15, -12, \ldots$

2.1. General Term of the Sequence

As established before, 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 the arithmetic sequence is given by: $a_n = a_1 + (n - 1) \times d$ Substituting the given values: $a_n = -21 + (n - 1) \times 3$ Simplifying: $a_n = 3n - 24$

2.2. Calculate the 100th Term

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

2.3. Verify if 0 is a Term in 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, specifically the 8th term.

2.4. Calculate the 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 we need to find $a_{15}$ using the general term formula: $a_{15} = 3(15) - 24 = 45 - 24 = 21$ Now, calculate the sum $S_{15}$: $S_{15} = \frac{15}{2} \times (-21 + 21) = \frac{15}{2} \times 0 = 0$ So, 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 related questions that might help you understand the topic further:

1. How do we identify if a sequence is arithmetic?
2. What happens to the sequence if the common difference is negative?
3. How would you calculate the sum of the first $n$ terms if the common difference were different?
4. Can you derive the general term for geometric sequences as well?
5. How do we know when a sequence term equals a specific value, like zero?
6. What role does the common difference play in determining the nature of the sequence?
7. How would the sum of the first $n$ terms change if the sequence started with a positive number?
8. Can we apply these formulas to sequences that are not arithmetic?

Tip: When verifying if a specific value is a term in the sequence, ensure that the resulting $n$ is a positive integer, as sequence terms are typically defined for positive indices.