For the following sequence, please fine the sum of the first 13 terms. Round to two decimal places. 128, 64, 32, 16

The given sequence is \( 128, 64, 32, 16, \ldots \), which is a geometric sequence because each term is obtained by multiplying the previous term by a constant ratio.

### Step 1: Identify the first term and the common ratio
- The first term (\(a\)) is \(128\).
- The common ratio (\(r\)) is obtained by dividing any term by the previous term:
\[
r = \frac{64}{128} = \frac{1}{2}.
\]

### Step 2: Formula for the sum of the first \(n\) terms of a geometric sequence
The formula for the sum of the first \(n\) terms of a geometric sequence is:
\[
S_n = a \frac{1 - r^n}{1 - r}, \quad \text{if } r \neq 1.
\]

### Step 3: Substitute the values
- \(a = 128\)
- \(r = \frac{1}{2}\)
- \(n = 13\)

\[
S_{13} = 128 \frac{1 - \left(\frac{1}{2}\right)^{13}}{1 - \frac{1}{2}}
\]

### Step 4: Simplify the denominator
\[
1 - \frac{1}{2} = \frac{1}{2}.
\]

\[
S_{13} = 128 \cdot 2 \cdot \left(1 - \left(\frac{1}{2}\right)^{13}\right)
\]

### Step 5: Calculate \( \left(\frac{1}{2}\right)^{13} \)
\[
\left(\frac{1}{2}\right)^{13} = \frac{1}{8192} \approx 0.000122.
\]

\[
S_{13} = 128 \cdot 2 \cdot \left(1 - 0.000122\right)
\]

\[
S_{13} = 128 \cdot 2 \cdot 0.999878
\]

\[
S_{13} \approx 128 \cdot 1.999756 = 255.97.
\]

### Final Answer:
The sum of the first 13 terms is approximately **255.97**.

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### Related Questions:
1. What is the sum of the first 20 terms of this sequence?
2. How does the sum behave as \(n \to \infty\)?
3. Can this formula be applied to sequences with \(r > 1\)?
4. How would the sum change if the sequence started at \(64\) instead of \(128\)?
5. What is the general term (\(a_n\)) of this sequence?

### Tip:
For geometric sequences, if \( |r| < 1 \), the sum of infinitely many terms converges to \( \frac{a}{1-r} \).

Learn how to calculate the sum of the first 13 terms of a geometric sequence starting with 128 and having a common ratio of 1/2. This problem involves key concepts in geometric sequences, including the sum formula and calculations. Suitable for students in Grades 8-10.

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