Answer
The sum of the first 113 terms of the series is approximately 18.
Solution
To find the sum of the first 113 terms of the series \( 6 + 4 + 3 + 2 + \frac{3}{2} + 0 + \frac{3}{4} + \ldots \), we first need to identify the pattern in the series.
### Step 1: Identify the series components
The series appears to be a combination of an arithmetic series and a geometric series. Let's break it down:
1. The first few terms are:
- \( 6 \)
- \( 4 \)
- \( 3 \)
- \( 2 \)
- \( \frac{3}{2} \)
- \( 0 \)
- \( \frac{3}{4} \)
2. The first part of the series (the first five terms) seems to decrease by a certain pattern:
- \( 6, 4, 3, 2, \frac{3}{2}, 0 \)
3. The last two terms \( \frac{3}{2} \) and \( 0 \) suggest a transition to a geometric series.
### Step 2: Analyze the arithmetic part
The first part of the series can be analyzed as follows:
- The first term is \( 6 \).
- The second term is \( 4 \) (decrease by \( 2 \)).
- The third term is \( 3 \) (decrease by \( 1 \)).
- The fourth term is \( 2 \) (decrease by \( 1 \)).
- The fifth term is \( \frac{3}{2} \) (decrease by \( \frac{1}{2} \)).
- The sixth term is \( 0 \) (decrease by \( \frac{3}{2} \)).
### Step 3: Identify the geometric part
After \( 0 \), the series seems to transition into a geometric series:
- The next term is \( \frac{3}{4} \).
### Step 4: Determine the pattern
The series can be split into two parts:
1. The arithmetic part: \( 6, 4, 3, 2, \frac{3}{2}, 0 \)
2. The geometric part: Starting from \( \frac{3}{4} \).
### Step 5: Calculate the sum of the first 113 terms
1. **Sum of the first 6 terms (arithmetic part)**:
\[
S_a = 6 + 4 + 3 + 2 + \frac{3}{2} + 0 = 15 + \frac{3}{2} = 16.5
\]
2. **Geometric series**:
The geometric series starts from \( \frac{3}{4} \) and continues. We need to find the number of terms in the geometric series.
The total number of terms is \( 113 \), and we have already accounted for \( 6 \) terms in the arithmetic part. Therefore, the number of terms in the geometric part is:
\[
113 - 6 = 107
\]
3. **Sum of the geometric series**:
The first term \( a = \frac{3}{4} \) and the common ratio \( r \) can be determined from the next term. Assuming the next term continues the pattern, we can find \( r \).
Let's assume the next term is \( \frac{3}{4} \cdot r \) and find \( r \) based on the pattern. If we assume the ratio is \( \frac{1}{2} \) (as it seems to halve), we can calculate the sum of the geometric series.
The sum of the first \( n \) terms of a geometric series is given by:
\[
S_g = a \frac{1 - r^n}{1 - r}
\]
where \( n = 107 \), \( a = \frac{3}{4} \), and \( r = \frac{1}{2} \).
Now, let's calculate the sum of the geometric series.
### Step 6: Calculate the sum of the geometric series
Using the formula:
\[
S_g = \frac{3}{4} \frac{1 - \left(\frac{1}{2}\right)^{107}}{1 - \frac{1}{2}} = \frac{3}{4} \cdot 2 \cdot \left(1 - \left(\frac{1}{2}\right)^{107}\right) = \frac{3}{2} \left(1 - \left(\frac{1}{2}\right)^{107}\right)
\]
### Step 7: Combine the sums
Finally, the total sum \( S \) of the first 113 terms is:
\[
S = S_a + S_g = 16.5 + \frac{3}{2} \left(1 - \left(\frac{1}{2}\right)^{107}\right)
\]
Now, let's calculate this value.
### Step 8: Final Calculation
Calculating \( S_g \):
\[
S_g = \frac{3}{2} \left(1 - \left(\frac{1}{2}\right)^{107}\right) \approx \frac{3}{2} \text{ (since } \left(\frac{1}{2}\right)^{107} \text{ is very small)}
\]
Thus,
\[
S \approx 16.5 + \frac{3}{2} = 16.5 + 1.5 = 18
\]
Therefore, the sum of the first 113 terms of the series is approximately \( 18 \).
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