sum of the first 15 terms of the sequence

To find the sum of the first 15 terms of a sequence, we need to know more about the sequence, such as whether it is arithmetic, geometric, or some other type. However, I'll cover the general formulas for both an arithmetic and a geometric sequence to guide you in solving the problem.

### 1. **Arithmetic Sequence**

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant difference, \(d\), to the previous term. The general formula for the \(n\)-th term of an arithmetic sequence is:

\[
a_n = a_1 + (n - 1)d
\]

where:
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.

The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence can be calculated using:

\[
S_n = \frac{n}{2} \times (2a_1 + (n - 1)d)
\]

For the first 15 terms (\(n = 15\)):

\[
S_{15} = \frac{15}{2} \times (2a_1 + 14d)
\]

### 2. **Geometric Sequence**

A geometric sequence is a sequence where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio, \(r\). The general formula for the \(n\)-th term of a geometric sequence is:

\[
a_n = a_1 \cdot r^{n-1}
\]

where:
- \(a_1\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.

The sum \(S_n\) of the first \(n\) terms of a geometric sequence is:

\[
S_n = a_1 \frac{1 - r^n}{1 - r} \quad \text{(if \(r \neq 1\))}
\]

For the first 15 terms (\(n = 15\)):

\[
S_{15} = a_1 \frac{1 - r^{15}}{1 - r}
\]

### **Conclusion**

To compute the sum of the first 15 terms, you need to know the type of sequence (arithmetic or geometric), as well as the first term \(a_1\), the common difference \(d\) (if arithmetic), or the common ratio \(r\) (if geometric). If you can provide these details, I can calculate the sum for you.

Would you like to provide more details, or do you have any questions?

### **Related Questions**

1. What is the sum of the first 10 terms of an arithmetic sequence with \(a_1 = 3\) and \(d = 5\)?
2. How do you derive the formula for the sum of the first \(n\) terms of a geometric sequence?
3. What is the sum of the first 20 terms of a geometric sequence with \(a_1 = 2\) and \(r = 3\)?
4. How do you find the sum of an infinite geometric series?
5. Can the formulas for arithmetic and geometric sequences be applied to non-integer values of \(n\)?

### **Tip:**  
Understanding the difference between arithmetic and geometric sequences is key to solving problems involving sums of terms. Remember that in an arithmetic sequence, each term is obtained by adding a constant, while in a geometric sequence, each term is obtained by multiplying by a constant.

Learn how to find the sum of the first 15 terms of a sequence using arithmetic or geometric sequence formulas. Understand the difference between arithmetic and geometric sequences and their respective sum formulas. Suitable for high school students and anyone interested in sequences and series summation.

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