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2.2.   Calcule o  valor  do centésimo termo
2.3.    Verifique e justifique se   0 é termo da sucessão.
2.4.  Calcule a soma dos quinze primeiros termos.

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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.

Explore how to calculate the 100th term, verify if 0 is a term, and find the sum of the first fifteen terms in an arithmetic sequence. This problem covers arithmetic progression, including formulas for the general term and sum of terms. Suitable for high school students.

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