`S_(n)` be the sum of n terms of the series `(8)/(5)+(16)/(65)+(24)/(325)+"......"`
The seveth term of the series is

Options

A. `(56)/(2505)`
B. `(56)/(6505)`
C. `(56)/(5185)`
D. `(107)/(9605)`

Explanation

To find the seventh term of the series given by \( \frac{8}{5} + \frac{16}{65} + \frac{24}{325} + \ldots \), we will first identify the general term of the series.

### Step 1: Identify the pattern in the series

The series can be expressed as:
- First term: \( \frac{8}{5} \)
- Second term: \( \frac{16}{65} \)
- Third term: \( \frac{24}{325} \)

We can see that the numerators are \( 8, 16, 24, \ldots \) which can be expressed as \( 8n \) for \( n = 1, 2, 3, \ldots \).

The denominators are \( 5, 65, 325, \ldots \). We can observe that:
- \( 5 = 5 \times 1 \)
- \( 65 = 5 \times 13 \)
- \( 325 = 5 \times 65 \)

It appears that the denominators can be expressed in a pattern related to powers of 5 and some other factor.

### Step 2: General term formulation

Let’s denote the \( n \)-th term of the series as \( T_n \). The general term can be expressed as:
\[ T_n = \frac{8n}{D_n} \]
where \( D_n \) is the denominator for the \( n \)-th term.

From the observed pattern, we can assume:
\[ D_n = 5 \times (n^2 + n) = 5n(n + 1) \]
Thus, the general term becomes:
\[ T_n = \frac{8n}{5n(n + 1)} = \frac{8}{5(n + 1)} \]

### Step 3: Calculate the seventh term

Now, we need to find the seventh term \( T_7 \):
\[ T_7 = \frac{8}{5(7 + 1)} = \frac{8}{5 \times 8} = \frac{8}{40} = \frac{1}{5} \]

### Final Answer

The seventh term of the series is:
\[ \boxed{\frac{1}{5}} \]