Find the sum to n terms of the series 3.8 +6.11 +9.14 + ...

Answer

To find the sum to n terms of the series \( \frac{3}{8} + \frac{6}{11} + \frac{9}{14} + \ldots \), we will first identify the pattern in the series and then derive the formula for the nth term.

### Step 1: Identify the Numerators and Denominators
The numerators of the series are:
- First term: 3
- Second term: 6
- Third term: 9

The pattern in the numerators is:
- \(3, 6, 9, \ldots\)

This can be expressed as:
\[ a_n = 3n \quad \text{(where \( n \) is the term number)} \]

The denominators of the series are:
- First term: 8
- Second term: 11
- Third term: 14

The pattern in the denominators is:
- \(8, 11, 14, \ldots\)

This can be expressed as:
\[ b_n = 8 + (n-1) \cdot 3 = 3n + 5 \]

### Step 2: Write the nth Term of the Series
Now we can write the nth term of the series as:
\[ T_n = \frac{a_n}{b_n} = \frac{3n}{3n + 5} \]

### Step 3: Find the Sum of n Terms
To find the sum of the first n terms, we need to sum up \( T_n \):
\[ S_n = \sum_{k=1}^{n} T_k = \sum_{k=1}^{n} \frac{3k}{3k + 5} \]

### Step 4: Simplifying the Sum
Calculating the sum directly can be complex, so we can use partial fractions or other techniques, but for simplicity, we can evaluate the sum numerically for small values of \( n \) to identify a pattern.

### Step 5: General Formula for the Sum
After evaluating the sum for small values of \( n \), we can derive a general formula, but this requires deeper analysis or computational assistance.

### Conclusion
The sum of the first n terms of the series can be expressed as:
\[ S_n = \sum_{k=1}^{n} \frac{3k}{3k + 5} \]
This requires further simplification or numerical evaluation for specific values of \( n \).